
Chapter 38 – Electromagnetic Induction
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Question 1:
A metallic loop is placed in a nonuniform magnetic field. Will an emf be induced in the loop?
Answer:
If the flux through the loop does not vary, there will be no induced emf. Because the magnetic field is nonuniform and does not change with time, there will be no change in the magnetic flux through the loop. Hence, no emf will be induced in the loop.
Question 2:
An inductor is connected to a battery through a switch. Explain why the emf induced in the inductor is much larger when the switch is opened as compared to the emf induced when the switch is closed.
Answer:
When we close the switch, the current takes some time to grow in the circuit. Due to this growth of current, the flux increases; hence, an emf is induced. On the other hand, when we open the switch, there is no path for the current to flow; hence, it suddenly drops to zero. This rate of decrease of current is much greater than the rate of growth of current when the switch is closed. So, when the switch is opened, the induced emf is more.
Question 3:
The coil of a movingcoil galvanometer keeps on oscillating for a long time if it is deflected and released. If the ends of the coil are connected together, the oscillation stops at once. Explain.
Answer:
When the ends of the coil are not connected, the coil acts as an inductor in which oscillations persist until the current decays slowly. When these ends are connected, the coil forms a close loop; hence, there is inductance across the ends and the coil does not behave like an inductor. Therefore, all oscillations stop at once.
Question 4:
A short magnet is moved along the axis of a conducting loop. Show that the loop repels the magnet if the magnet is approaching the loop and attracts the magnet if it is going away from the loop.
Answer:
Consider the above situation in which a magnet is moved towards a conducting circular loop. The north pole of the magnet faces the loop. As the magnet comes closer to the loop, the magnetic field increases; hence, flux through the loop increases. According to Lenz’s law, the direction of induced current is such that it opposes the magnetic field that has induced it. Thus, the induced current produces a magnetic field in the direction opposite to the original field; hence, the loop repels the magnet.
On the other hand, when the magnet is going away from the loop, the magnetic field decreases. Hence, flux through the loop decreases. According to Lenz’s law, the induced current produces a magnetic field in the opposite direction of the original field; hence, the loop attracts the magnet.
Question 5:
Two circular loops are placed coaxially but separated by a distance. A battery is suddenly connected to one of the loops establishing a current in it. Will there be a current induced in the other loop? If yes, when does the current start and when does it end? Do the loops attract each other or do they repel?
Answer:
Consider loops A and B placed coaxially as shown above. Let the direction of the current in loop A be clockwise when the battery is connected to it. According to the righthand screw rule, the direction of the magnetic field due to this current will be towards left, as seen from the side of B. Due to a sudden flux through loop B, a current will be induced in it. It will only be induced for a moment when the current suddenly jumps from zero to a constant value. After it has attained a constant value, there will be no induced current. Now, according to Lenz’s law, the direction of the induced current in loop B will be such that it will oppose the magnetic field due to loop A. Hence, a current will be induced in anticlockwise direction in loop B. The induced current will flow in loop B as soon as the current grows in loop A and will end when the current through loop A becomes zero. Because the directions of the currents in the loops are opposite, they will repel each other.
Question 6:
The battery discussed in the previous question is suddenly disconnected. Is a current induced in the other loop? If yes, when does it start and when does it end? Do the loops attract each other or repel?
Answer:
When the battery is suddenly disconnected, a current is induced in loop B due to a sudden change in the flux through it. It is only induced for a moment when the current suddenly falls to zero. There is no induced current after it has fallen to zero. According to Lenz’s law, the induced current is such that it increases the decreasing magnetic field. So, if the current in loop A is in clockwise direction, the induced current in loop B will also be in clockwise direction. Hence, the two loops will attract each other.
Question 7:
If the magnetic field outside a copper box is suddenly changed, what happens to the magnetic field inside the box? Such lowresistivity metals are used to form enclosures which shield objects inside them against varying magnetic fields.
Answer:
The varying magnetic field induces eddy currents on the walls of the copper box. There is a magnetic field due to the induced eddy currents, that is in opposite direction. As copper has good conductivity, thus the magnetic field due to the eddy currents will be strong. The magnetic field induced due to eddy currents in the copper walls cancel the original magnetic field. Thus, magnetic field does not penetrate the enclosure made of copper. The magnetic field inside the box remains zero. This is how a copper box protects the inside material from varying magnetic fields.
Question 8:
Metallic (nonferromagnetic) and nonmetallic particles in a solid waste may be separated as follows. The waste is allowed to slide down an incline over permanent magnets. The metallic particles slow down as compared to the nonmetallic ones and hence are separated. Discuss the role of eddy currents in the process.
Answer:
When solid waste is allowed to slide over a permanent magnet, an emf is induced in metallic particles. This is because magnetic flux linked with the particles changes in this case. According to Lenz’s law this induced emf opposes its cause i.e. downward motion along the inclined plane of the permanent magnet. On the other hand, nonmetallic or insulating particles are free from such effects. As a result, the metallic particles slow down and hence get separated from the waste (or nonmetallic particles).
Question 9:
A pivoted aluminium bar falls much more slowly through a small region containing a magnetic field than a similar bar of an insulating material. Explain.
Answer:
An aluminium bar falls slowly through a small region containing a magnetic field because of the induced eddy currents (or induced emf) in it. According to Lenz’s law this induced eddy current oppose its cause (its motion). Hence, it slows down while falling through a region containing a magnetic field. On the other hand, nonmetallic or insulating materials are free from such effects.
Question 10:
A metallic bob A oscillates through the space between the poles of an electromagnet (figure). The oscillations are more quickly damped when the circuit is on, as compared to the case when the circuit is off. Explain.
Figure
Answer:
When the circuit is on, eddy currents are produced on the surface of the metallic bob. Due to these eddy currents, thermal energy is generated in it. This thermal energy comes at the cost of the kinetic energy of the bob; hence, oscillations are more quickly damped when the circuit is on compared to when the circuit is off.
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Question 11:
Two circular loops are placed with their centres separated by a fixed distance. How would you orient the loops to have (a) the largest mutual inductance (b) the smallest mutual inductance?
Answer:
(a) For the largest mutual inductance, the two loops should be placed coaxially. In this case, flux through a loop due to another loop is the largest; hence, mutual inductance is the largest.
(b) For the smallest mutual inductance, the two loops should be placed such that their axes are perpendicular to each other. In this case, flux through a loop due to another loop is the smallest (zero); hence, mutual inductance is the smallest.
Question 12:
Consider the selfinductance per unit length of a solenoid at its centre and that near its ends. Which of the two is greater?
Answer:
Selfinductance per unit length of solenoid is given as,
Ll=μ0n2AFrom the above equation we can see that, self inductance per unit length will depend on the permeability of free space (
μ0), number of turns per unit length (n) and area of the crosssection of the solenoid (A). All the above factors are constant at the centre and near any end of the solenoid therefore self inductance at both the points will be same.
Question 13:
Consider the energy density in a solenoid at its centre and that near its ends. Which of the two is greater?
Answer:
In a solenoid energy is stored in the form of magnetic field. If a constant current is flowing from a solenoid then magnetic field inside the solenoid is uniform. Therefore, energy per unit volume (or energy density) in the magnetic field inside the solenoid is constant.
The energy per unit volume in the magnetic field is given as,
u=B2μ0, where B is uniform magnetic field inside the solenoid.
Therefore, energy density all points inside a solenoid is same.
Question 1:
A rod of length l rotates with a small but uniform angular velocity ω about its perpendicular bisector. A uniform magnetic field B exists parallel to the axis of rotation. The potential difference between the centre of the rod and an end is
(a) zero
(b)
18ωBl2(c)
12ωBl2(d) Bωl^{2}.
Answer:
(b)
18ωBl2
Let us consider a small element dx at a distance x from the centre of the rod rotating with angular velocity ω about its perpendicular bisector. The emf induced in the rod because of this small element is given by
de=Bvl=BωxdxThe emf induced across the centre and end of the rod is given by
∫de=∫0l/2Bωxdx⇒E=Bωx220l/2⇒E=18Bωl2
Question 2:
A rod of length l rotates with a uniform angular velocity ω about its perpendicular bisector. A uniform magnetic field B exists parallel to the axis of rotation. The potential difference between the two ends of the rod is
(a) zero
(b)
12Blω2(c) Blω^{2}
(d) 2Blω^{2}.
Answer:
(a) zero
Let us consider a small element dx at a distance x from the centre of the rod rotating with angular velocity ω about its perpendicular bisector. The emf induced in the small element of the rod because of its motion is given by
de=BωxdxThe emf induced between the centre of the rod and one of its end is given by
∫de=∫0lBωxdx⇒e=Bωx220l/2⇒e=18Bωl2The emf at both ends is the same. So, the potential difference between the two ends is zero.
Question 3:
Consider the situation shown in figure. If the switch is closed and after some time it is opened again, the closed loop will show
(a) an anticlockwise currentpulse
(b) a clockwise currentpulse
(c) an anticlockwise currentpulse and then a clockwise currentpulse
(d) a clockwise currentpulse and then an anticlockwise currentpulse
Figure
Answer:
(d) a clockwise currentpulse and then an anticlockwise currentpulse
When the switch is closed, the current will flow in downward direction in part AB of the circuit nearest to the closed loop.
Due to current in wire AB, a magnetic field will be produced in the loop. This magnetic field due to increasing current will be the cause of the induced current in the closed loop. According to Lenz’s law, the induced current is such that it opposes the increase in the magnetic field that induces it. So, the induced current will be in clockwise direction opposing the increase in the magnetic field in upward direction.
Similarly, when the circuit is opened, the current will suddenly fall in the circuit, leading to decrease in the magnetic field in the loop. Again, according to Lenz’s law, the induced current is such that it opposes the decrease in the magnetic field. So, the induced current will be in anticlockwise direction, opposing the decrease in the magnetic field in upward direction.
Question 4:
Solve the previous question if the closed loop is completely enclosed in the circuit containing the switch.
Answer:
(c) an anticlockwise currentpulse and then a clockwise currentpulse
According to Lenz’s law, the induced current in the loop will be such that it opposes the increase in the magnetic field due to current flow in the circuit. Therefore, the direction of the induced current when the switch is closed is anticlockwise.
Similarly, when the switch is open, there is a sudden fall in the current, leading to decrease in the magnetic field at the centre of the loop. According to Lenz’s law, the induced current in the loop is such that it opposes the decrease in the magnetic field. Therefore, the direction of the induced current when the switch is open is clockwise.
Question 5:
A bar magnet is released from rest along the axis of a very long, vertical copper tube. After some time the magnet
(a) will stop in the tube
(b) will move with almost contant speed
(c) will move with an acceleration g
(d) will oscillate.
Answer:
(b) will move with almost constant speed
As the magnet is moving under gravity, the flux linked with the copper tube will change because of the motion of the magnet. This will produce eddy currents in the body of the copper tube. According to Lenz’s law, these induced currents oppose the fall of the magnet. So, the magnet will experience a retarding force. This force will continuously increase with increasing velocity of the magnet till it becomes equal to the force of gravity. After this, the net force on the magnet will become zero. Hence, the magnet will attain a constant speed.
Question 6:
Figure shows a horizontal solenoid connected to a battery and a switch. A copper ring is placed on a frictionless track, the axis of the ring being along the axis of the solenoid. As the switch is closed, the ring will
(a) remain stationary
(b) move towards the solenoid
(c) move away from the solenoid
(d) move towards the solenoid or away from it depending on which terminal (positive or negative) of the battery is connected to the left end of the solenoid.
Figure
Answer:
(c) move away from the solenoid
For the circuit,
E=LdidtThe current will increase in the solenoid, flowing in clockwise direction in the circuit. Due to this increased current, the flux linked with the copper ring with increase with time, causing an induced current. This induced current will oppose the cause producing it. Hence, the current in the copper ring will be in anticlockwise direction. Now, because the directions of currents in the solenoid and ring are opposite, the ring will be repelled and hence will move away from the solenoid.
Question 7:
Consider the following statements:
(A) An emf can be induced by moving a conductor in a magnetic field.
(B) An emf can be induced by changing the magnetic field.
(a) Both A and B are true.
(b) A is true but B is false.
(c) B is true but A is false.
(d) Both A and B are false.
Answer:
(a) Both A and B are true.
Statement A is true, as an emf can be induced by moving a conductor with some velocity v in a magnetic field B. It is given by
e=BvlStatement B is true, as an emf can be induced by changing the magnetic field that causes the change in flux Ï• through a conductor or a loop. It is given by
e=dϕdt
Question 8:
Consider the situation shown in figure. The wire AB is slid on the fixed rails with a constant velocity. If the wire AB is replaced by a semicircular wire, the magnitude of the induced current will
(a) increase
(b) remain the same
(c) decrease
(d) increase or decrease depending on whether the semicircle bulges towards the resistance or away from it.
Figure
Answer:
The induced emf across ends A and B is given by
E=BvlThis induced emf will serve as a voltage source for the current to flow across resistor R, as shown in the figure. The direction of the current is given by Lenz’s law and it is anticlockwise.
i=BvlRIf the wire is replaced by a semicircular wire, the induced current will remain the same, as it depends on the length of the wire and not on its shape (when B, v and R are kept constant).
Question 9:
Figure shows a conducting loop being pulled out of a magnetic field with a speed v. Which of the four plots shown in figure (b) may represent the power delivered by the pulling agent as a function of the speed v?
Figure
Answer:
(b) b
The emf developed across the ends of the loop is given by
e=BvlIf R is the resistance of the loop, then the power delivered to the loop is given by
P=e2R=B2v2l2R⇒P∝v2This relation is best represented by plot b in the figure.
Question 10:
Two circular loops of equal radii are placed coaxially at some separation. The first is cut and a battery is inserted in between to drive a current in it. The current changes slightly because of the variation in resistance with temperature. During this period, the two loops
(a) attract each other
(b) repel each other
(c) do not exert any force on each other
(d) attract or repel each other depending on the sense of the current.
Answer:
Consider loops A and B placed coaxially as above. Let the direction of the current in loop A be clockwise when a battery is connected to it. According to the righthand screw rule, the direction of the magnetic field due to this current will be towards left. Now, the current through this loop will decrease with time due to increase in resistance with temperature. So, the magnetic field due to this current will also decrease with time. This changing current will induce current in loop B. Now, according to Lenz’s law, the direction of the induced current in loop B will be such that it will oppose the decrease in the magnetic field due to loop A. Hence, current will be induced in clockwise direction in loop B. Also, because the direction of the currents in the loops is the same, they will attract each other.
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Question 11:
A small, conducting circular loop is placed inside a long solenoid carrying a current. The plane of the loop contains the axis of the solenoid. If the current in the solenoid is varied, the current induced in the loop is
(a) clockwise
(b) anticlockwise
(c) zero
(d) clockwise or anticlockwise depending on whether the resistance is increased or decreased.
Answer:
(c) zero
The magnetic field inside the solenoid is parallel to its axis. If the plane of the loop contains the axis of the solenoid, then the angle between the area vector of the circular loop and the magnetic field is zero. Thus, the flux through the circular loop is given by
ϕ=BAcosθ=BAcos0°=BAHere,
B = Magnetic field due to the solenoid
A = Area of the circular loop
θ = Angle between the magnetic field and the area vector
Now, the induced emf is given by
e=dϕdt∵φ=BA=constant∴e=0We can see that the induced emf does not depend on the varying current through the solenoid and is zero for constant flux through the loop. Because there is no induced emf, no current is induced in the loop.
Question 12:
A conducting square loop of side l and resistance R moves in its plane with a uniform velocity v perpendicular to one of its sides. A uniform and constant magnetic field B exists along the perpendicular to the plane of the loop as shown in figure. The current induced in the loop is
(a) Blv/R clockwise
(b) Blv/R anticlockwise
(c) 2Blv/R anticlockwise
(d) zero.
Figure
Answer:
(d) zero
Figure (a) shows the square loop moving in its plane with a uniform velocity v.
Figure (b) shows the equivalent circuit.
The induced emf across ends AB and CD is given by
E=BvlOn applying KVL in the equivalent circuit, we get
EE+iR=0⇒i=0No current will be induced in the circuit due to zero potential difference between the closed ends.
Question 1:
A bar magnet is moved along the axis of a copper ring placed far away from the magnet. Looking from the side of the magnet, an anticlockwise current is found to be induced in the ring. Which of the following may be true?
(a) The south pole faces the ring and the magnet moves towards it.
(b) The north pole faces the ring and the magnet moves towards it.
(c) The south pole faces the ring and the magnet moves away from it.
(d) The north pole faces the ring and the magnet moves away from it.
Answer:
(b) The north pole faces the ring and the magnet moves towards it.
(c) The south pole faces the ring and the magnet moves away from it.
It can be observed that the induced current is in anticlockwise direction. So, the magnetic field induced in the copper ring is towards the observer.
According to Lenz’s law, the current induced in a circuit due to a change in the magnetic flux is in such direction so as to oppose the change in flux.
Two cases are possible:
(1) The magnetic flux is increasing in the direction from the observer to the circular coil.
(2) The magnetic flux is decreasing in the direction from the coil to the observer.
So, from the above mentioned points, the following conclusions can be made:
1. The south pole faces the ring and the magnet moves away from it.
2. The north pole faces the ring and the magnet moves towards it.
Question 2:
A conducting rod is moved with a constant velocity v in a magnetic field. A potential difference appears across the two ends
(a) if
v →∥l →(b) if
v →∥B →(c) if
l →∥B →(d) none of these.
Answer:
(d) none of these
The potential difference across the two ends is given by
e=BvlIt is nonzero only
(i) if the rod is moving in the direction perpendicular to the magnetic field (
v→⊥B→)
(ii) if the velocity of the rod is in the direction perpendicular to the length of the rod
(
v→⊥l →)
(iii) if the magnetic field is perpendicular to the length of the rod
l→⊥B→Thus, none of the above conditions is satisfied in the alternatives given.
Question 3:
A conducting loop is placed in a uniform magnetic field with its plane perpendicular to the field. An emf is induced in the loop if
(a) it is translated
(b) it is rotated about its axis
(c) it is rotated about a diameter
(d) it is deformed.
Answer:
(c) it is rotated about a diameter
(d) it is deformed
When translated or rotated about its axis, the magnetic flux through the loop does not change; hence, no emf is induced in the loop.
When rotated about a diameter, the magnetic flux through the loop changes and emf is induced. On deforming the loop, the area of the loop inside the magnetic field changes, thereby changing the magnetic flux. Due to the change in the flux, emf is induced in the loop.
Question 4:
A metal sheet is placed in front of a strong magnetic pole. A force is needed to
(a) hold the sheet there if the metal is magnetic
(b) hold the sheet there if the metal is nonmagnetic
(c) move the sheet away from the pole with uniform velocity if the metal is magnetic
(d) move the sheet away from the pole with uniform velocity if the metal is nonmagnetic.
Neglect any effect of paramagnetism, diamagnetism and gravity.
Answer:
(a) hold the sheet there if the metal is magnetic
(c) move the sheet away from the pole with uniform velocity if the metal is magnetic
(d) move the sheet away from the pole with uniform velocity if the metal is nonmagnetic
The strong magnetic pole will attract the magnet, so a force is needed to hold the sheet there if the metal is magnetic.
If we move the metal sheet (magnetic or nonmagnetic) away from the pole, eddy currents are induced in the sheet. Because of eddy currents, thermal energy is produced in it. This energy comes at the cost of the kinetic energy of the plate; thus, the plate slows down. So, a force is needed to move the sheet away from the pole with uniform velocity.
Question 5:
A constant current i is maintained in a solenoid. Which of the following quantities will increase if an iron rod is inserted in the solenoid along its axis?
(a) magnetic field at the centre
(b) magnetic flux linked with the solenoid
(c) selfinductance of the solenoid
(d) rate of Joule heating.
Answer:
(a) magnetic field at the centre
(b) magnetic flux linked with the solenoid
(c) selfinductance of the solenoid
Iron rod has high permeability. When it is inserted inside a solenoid the magnetic field inside the solenoid increases. As magnetic field increases inside the solenoid thus the magnetic flux also increases. The Selfinductance (L) of the coil is directly proportional to the permeability of the material inside the solenoid. As the permeability inside the coil increases. Therefore, the selfinductance will also increase.
Question 6:
Two solenoids have identical geometrical construction but one is made of thick wire and the other of thin wire. Which of the following quantities are different for the two solenoids?
(a) selfinductance
(b) rate of Joule heating if the same current goes through them
(c) magnetic field energy if the same current goes through them
(d) time constant if one solenoid is connected to one battery and the other is connected to another battery.
Answer:
(b) rate of Joule heating if the same current goes through them
(d) time constant if one solenoid is connected to one battery and the other is connected to another battery
Because the solenoids are identical, their selfinductance will be the same.
Resistance of a wire is given by
R=ρlAHere,
l = Length of the wire
A = Area of cross section of the wire
ρ = Resistivity of the wire
Because ρ and l are the same for both wires, the thick wire will have greater area of cross section and hence less resistance than the thin wire.
⇒Rthick <RthinThe time constant for a solenoid is given by
τ=LR
∴τthick>τthinThus, time constants of the solenoids would be different if one solenoid is connected to one battery and the other is connected to another battery.
Also, because the selfinductance of the solenoids is the same and the same current flows through them, the magnetic field energy given by
12Li2will be the same.
Power dissipated as heat is given by
P=i2Ri is the same for both solenoids.
∴Pthick<PthinBecause the resistance of the coils are different, the rate of Joule heating will be different for the coils if the same current goes through them.
Question 7:
An LR circuit with a battery is connected at t = 0. Which of the following quantities is not zero just after the connection?
(a) Current in the circuit
(b) Magnetic field energy in the inductor
(c) Power delivered by the battery
(d) Emf induced in the inductor
Answer:
(d) Emf induced in the inductor
At time t = 0, the current in the LR circuit is zero. The magnetic field energy is given by
U=12Li2, as the current is zero the magnetic field energy will also be zero. Thus, the power delivered by the battery will also be zero. As, the LR circuit is connected to the battery at t = 0, at this time the current is on the verge to start growing in the circuit. So, there will be an induced emf in the inductor at the same time to oppose this growing current.
Question 8:
A rod AB moves with a uniform velocity v in a uniform magnetic field as shown in figure.
(a) The rod becomes electrically charged.
(b) The end A becomes positively charged.
(c) The end B becomes positively charged.
(d) The rod becomes hot because of Joule heating.
Figure
Answer:
(b) The end A becomes positively charged.
Due to electromagnetic induction, emf e is induced across the ends of the rod. This induced emf is given by
e=BvlThe direction of this induced emf is from A to B, that is, A is at the higher potential and B is at the lower potential. This is because the magnetic field exerts a force equal to
qvBon each free electron where q is
1.6 × 10^{16} C. The force is towards AB by Fleming’s lefthand rule; hence, negatively charged electrons move towards the end B and get accumulated near it. So, a negative charge appears at B and a positive charge appears at A.
Question 9:
L, C and R represent the physical quantities inductance, capacitance and resistance respectively. Which of the following combinations have dimensions of frequency?
(a)
1RC(b)
RL(c)
1LC(d) C/L.
Answer:
(a)
1RC(b)
RL(c)
1LCThe time constant of the RC circuit is given by
τ=RCOn taking the reciprocal of the above relation, we get
f1=1RC …(1)
f_{1} will have the dimensions of the frequency.
The time constant of the LR circuit is given by
τ=LROn taking the reciprocal of the above relation, we get
f2=RL …(2)
f_{2} will have the dimensions of the frequency.
On multiplying eq. (1) and (2), we get
f1f2=1LC⇒f1f2=1LCThus,
f1f2will have the dimensions of the frequency.
Question 10:
The switches in figure (a) and (b) are closed at t = 0 and reopened after a long time at t = t_{0}.
Figure
(a) The charge on C just after t = 0 is εC.
(b) The charge on C long after t = 0 is εC.
(c) The current in L just before t = t_{0} is ε/R.
(d) The current in L long after t = t_{0} is ε/R.
Answer:
(b) The charge on C long after t = 0 is εC.
(d) The current in L long after t = t_{0} is ε/R.
The charge on the capacitor at time ”t” after connecting it with a battery is given by,
Q=Cε1et/RCJust after t = 0, the charge on the capacitor will be
Q=Cε1e0=0For a long after time,
t→∞Thus, the charge on the capacitor will be
Q=Cε1e∞⇒Q=Cε10=CεThe current in the inductor at time ”t” after closing the switch is given by
I=VbR1etR/LJust before the time t_{0}, current through the inductor is given by
I=VbR1et0R/LIt is given that the time t_{0} is very long.
∴
t0→∞
I=εR1e∞=εRWhen the switch is opened, the current through the inductor after a long time will become zero.
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Question 1:
Calculate the dimensions of (a)
∫E→.dl,→(b) vBl and (c)
dΦBdt. The symbols have their usual meaning.
Answer:
(a) The quantity
∫E.dl can also be written as:
∫E.dl = V (V = Voltage)
Unit of voltage is J/C.
Voltage can be written as:
Voltage=EnergyChargeDimensions of energy = [ML^{2}T^{2}]
Dimensions of charge = [IT]
Thus, the dimensions of voltage can be written as:
[ML^{2}T^{2}] ×[IT]^{−1} = [ML^{2}I^{−1}T^{−3}]
(b) The quantity vBl is the product of quantities v, B and L.
Dimensions of velocity v = [LT^{−1}]
Dimensions of length l = [L]
The dimensions of magnetic field B can be found using the following formula:
B=FqvDimensions of force F = [MLT^{−2}^{]}
Dimensions of charge q = [IT]
Dimensions of velocity = [LT^{−1}]
The dimensions of a magnetic field can be written as:
MI^{−1}T^{−2}
∴ Dimensions of vBl = [LT^{−1}] × [MI^{−1}T^{−2}] × [L]= [ML^{2}I^{−1}T^{−3}]
(c) The quantity
dϕdtis equal to the emf induced; thus, its dimensions are the same as that of the voltage.
Voltage can be written as:
Voltage=EnergyChargeDimensions of energy = [ML^{2}T^{2}]
Dimensions of charge = [IT]
The dimensions of voltage can be written as:
[ML^{2}T^{2}] ×[IT]^{−1} = [ML^{2}I^{−1}T^{−3}]
∴ Dimensions of
dϕdt= [ML^{2}I^{−1}T^{−3}]
Question 2:
The flux of magnetic field through a closed conducting loop changes with time according to the equation, Φ = at^{2} + bt + c. (a) Write the SI units of a, b and c. (b) If the magnitudes of a, b and c are 0.20, 0.40 and 0.60 respectively, find the induced emf at t = 2 s.
Answer:
According to the principle of homogeneity of dimensions, the dimensions of each term on both the sides of a correct equation must be the same.
Now,
Ï• = at^{2} + bt + c
(a) The dimensions of the quantities at^{2}, bt, c and Ï• must be the same.
Thus, the units of the quantities are as follows:
a=ϕt2=ϕ/tt=Voltsb=ϕt=Voltc=ϕ=Weber(b) The emf is written as:
E=dϕdt= 2at + b = 2 × 0.2 × 2 + 0.4 (âˆµ a = 0.2, b = 0.4 and c = 0.6)
On substituting t = 2 s, we get
E = 0.8 + 0.4
= 1.2 V
Question 3:
(a) The magnetic field in a region varies as shown in figure. Calculate the average induced emf in a conducting loop of area 2.0 × 10^{−3} m^{2} placed perpendicular to the field in each of the 10 ms intervals shown. (b) In which intervals is the emf not constant? Neglect the behaviour near the ends of 10 ms intervals.
Figure
Answer:
Given:
Area of the loop = 2.0 × 10^{−3} m^{2}
The following conclusions can be made from the graph given above:
The magnetic flux at point O is 0.
The magnetic flux at point A is given by
Ï•_{2} = B.A = 0.01 × 2 × 10^{−3}
= 2 × 10^{−5} [âˆµ Ï•_{1} = 0]
The change in the magnetic flux in 10 ms is given by
∆Ï• = 2 × 10^{−5}
The emf induced is given by
e=∆ϕ ∆t=2×105010×103=2 mVThe magnetic flux at point B is given by
Ï•_{3} = B.A = 0.03 × 2 × 10^{−3}
= 6 × 10^{−5}
The change in the magnetic flux in 10 ms is given by
∆Ï• = 6 × 10^{−5} − 2 × 10^{−5} = 4 × 10^{−5}
The emf induced is given by
e=∆ϕ ∆t=4 mVThe magnetic flux at point C is given by
Ï•_{4} = B.A = 0.01 × 2 × 10^{−3}
= 2 × 10^{−5}
The change in the magnetic flux in 10 ms is given by
∆Ï• = (2 × 10^{−5} − 6 × 10^{−5} ) = − 4 × 10^{−5}
The emf induced is given by
e=∆ϕ∆t=4 mVThe magnetic flux at point D is given by
Ï•_{5} = B.A = 0
The change in the magnetic flux in 10 ms is given by
∆Ï• = 0 − 2 × 10^{−5}
The emf induced is given by
e=∆ϕ∆t=(2)×10510×103=2 mV(b) Emf is not constant in the intervals 10 msâ€’20 ms and 20 msâ€’30 ms.
Question 4:
A conducting circular loop having a radius of 5.0 cm, is placed perpendicular to a magnetic field of 0.50 T. It is removed from the field in 0.50 s. Find the average emf produced in the loop during this time.
Answer:
Given:
Magnetic field intensity, B = 0.50 T
Radius of the loop, r = 5.0 cm = 5 × 10^{−2} m
∴ Area of the loop, A =
πr2Initial magnetic flux in the loop, Ï•_{1} = B × A
Ï•_{1} = 0.5 ×
π(5 × 10^{−2})^{2} = 125
π× 10^{−5}
As the loop is removed from the magnetic field, magnetic flux (Ï•_{2}) = 0.
Induced emf ε is given by
ε=ϕ1ϕ2t =125π×1055×101 =25π×104 = 25 × 3.14 × 10^{−4}
= 78.5 × 10^{−4} V = 7.8 × 10^{−3} V
Question 5:
A conducting circular loop of area 1 mm^{2} is placed coplanarly with a long, straight wire at a distance of 20 cm from it. The straight wire carries an electric current which changes from 10 A to zero in 0.1 s. Find the average emf induced in the loop in 0.1 s.
Answer:
Given:
Area of the loop, A = 1 mm^{2}
Current through the wire, i = 10 A
Separation between the wire and the loop, d = 20 cm
Time, dt = 0.1 s
The average emf induced in the loop is given by
e=dϕdt =BAdt=μ0 i2πd×Adt =4π×107×102π×2×101×1061×101 =1×1010 V
Question 6:
A squareshaped copper coil has edges of length 50 cm and contains 50 turns. It is placed perpendicular to a 1.0 T magnetic field. It is removed from the magnetic field in 0.25 s and restored in its original place in the next 0.25 s. Find the magnitude of the average emf induced in the loop during (a) its removal, (b) its restoration and (c) its motion.
Answer:
(a) When the coil is removed from the magnetic field:
Initial magnetic flux through the coil, Ï•_{1} = BA
∴ Ï•_{1} = 50 × 0.5 × 0.5 Tm^{2}
= 12.5 Tm^{2}
Now,
Initial magnetic flux through the coil, Ï•_{2} = 0
Time taken, t = 0.25 s
The average emf induced is given by
e=∆ϕ∆t=ϕ1ϕ2dt =12.500.25=125×10125×102=50 V
(b) When the coil is taken back to its original position:
Initial magnetic flux through the coil, Ï•_{1} = 0
Initial magnetic flux through the coil, Ï•_{2} = 12.5 Tm^{2}
Time taken, t = 0.25 s
The average emf induced is given by
e=12.500.25=50 V(c) When the coil is moving outside the magnetic field:
Initial magnetic flux, Ï•_{1} = 0
Final magnetic flux, Ï•_{2} = 0
Because there is no change in the magnetic flux, no emf is induced.
Question 7:
Suppose the resistance of the coil in the previous problem is 25â„¦. Assume that the coil moves with uniform velocity during its removal and restoration. Find the thermal energy developed in the coil during (a) its removal, (b) its restoration and (c) its motion.
Answer:
Given:
Resistance of the coil, R = 25 Ω
(a) During the removal the emf induced in the coil,
e = 50 V
time taken, t = 0.25 s
current in the coil,
i=eR=2AThus, the thermal energy developed is given by
H = I^{2}RT
= 4 × 25 × 0.25 = 25 J
(b) During the restoration of the coil,
emf induced in it, e = 50 V
time taken, t = 0.25 s
current in the coil,
i=eR=2AThus, the thermal energy developed is given by
H = i^{2}RT = 25 J
(c) We know that energy is a scalar quantity. Also, the net thermal energy is the algebraic sum of the two energies calculated.
∴ Net thermal energy developed
= 25 J + 25 J = 50 J
Question 8:
A conducting loop of area 5.0 cm^{2} is placed in a magnetic field which varies sinusoidally with time as B = B_{0} sin ωt where B_{0} = 0.20 T and ω = 300 s^{−1}. The normal to the coil makes an angle of 60° with the field. Find (a) the maximum emf induced in the coil, (b) the emf induced at τ = (π/900)s and (c) the emf induced at t = (π/600) s.
Answer:
Given:
Area of the coil, A = 5 cm^{2} = 5 × 10^{−4} m^{2}
The magnetic field at time t is given by
B = B_{0} sin ωt = 0.2 sin (300t)
Angle of the normal of the coil with the magnetic field, θ = 60°
(a) The emf induced in the coil is given by
e=dθ dt=ddt(BA cos θ) =ddtB0 sin ωt×5×104×1/2 =B0×52×104ddt(sin ωt) =B052104 ωcos ωt =0.2×52×300×104×cos ωt =15×103cost ωtThe induced emf becomes maximum when cos ωt becomes maximum, that is, 1.
Thus, the maximum value of the induced emf is given by
emax=15×103=0.015 V(b) The induced emf at t =
π900 sis given by
e = 15 × 10^{−3} × cos ωt
= 15 × 10^{−3} × cos
300×π900 = 15 × 10^{−3} ×
12
=0.0152=0.0075=7.5×103 V(c) The induced emf at t =
π600 sis given by
e = 15 × 10^{−3} × cos
300×π600 = 15 × 10^{−3} × 0 = 0 V
Question 9:
Figure shows a conducting square loop placed parallel to the polefaces of a ring magnet. The polefaces have an area of 1 cm^{2} each and the field between the poles is 0.10 T. The wires making the loop are all outside the magnetic field. If the magnet is removed in 1.0 s, what is the average emf induced in the loop?
Figure
Answer:
It is given that the magnitude of the magnetic field is 0.10 T and it is perpendicular to the area of the loop.
Also,
Area of the loop, A = 1 cm^{2} = 10^{−4} m
Time taken to remove the magnet completely, T = 2 s
Initial magnetic flux, Ï• =
B→.A→= BA cos(0) = 10^{−1} × 10^{−4} × 1 = 10^{−}^{5}
Now, the induced emf in the magnetic field is given by
e=∆ϕ∆t=10501=105=10 μV
Question 10:
A conducting square loop having edges of length 2.0 cm is rotated through 180° about a diagonal in 0.20 s. A magnetic field B exists in the region which is perpendicular to the loop in its initial position. If the average induced emf during the rotation is 20 mV, find the magnitude of the magnetic field.
Answer:
Given:
Induced emf, e = 20 mV = 20 × 10^{−3} V
Area of the loop, A = (2 × 10^{−2})^{2} = 4 × 10^{−4} m^{2}
Time taken to rotate the loop, Δt = 0.2 s
The average induced emf is given by
e=∆ϕ∆t=ϕiϕftϕ=B→.A→=BAcosθϕi=B(4×104)cos0=B(4×104)ϕf=B(4×104)cos180o=B(4×104)e=B(4×104)–B(4×104)0.220×103=8B×1042×101
⇒20×103=4×B×103⇒B=20×1034×103=5 T
Question 11:
A conducting loop of facearea A and resistance R is placed perpendicular to a magnetic field B. The loop is withdrawn completely from the field. Find the charge which flows through any crosssection of the wire in the process. Note that it is independent of the shape of the loop as well as the way it is withdrawn.
Answer:
The magnetic flux through the coil is given by
Ï• = B.A = BA cos 0° = BA
It is given that the loop is withdrawn from the magnetic field.
∴ Final flux = 0
The average induced emf is given by
e=∆ϕ∆t=BA0t=BAtThe current in the loop is given by
i=eR=BAtRThe charge flowing through the area of the cross section of the wire is given by
q=it=BAR
Question 12:
radiuA long solenoid ofs 2 cm has 100 turns/cm and carries a current of 5 A. A coil of radius 1 cm having 100 turns and a total resistance of 20 â„¦ is placed inside the solenoid coaxially. The coil is connected to a galvanometer. If the current in the solenoid is reversed in direction, find the charge flown through the galvanometer.
Answer:
Given:
Radius of the solenoid, r = 2 cm = 2 × 10^{−2} m
Number of turns per centimetre, n = 100 = 10000 turns/m
Current flowing through the coil, i = 5 A
The magnetic field through the solenoid is given by
B = μ_{0}ni = 4π × 10^{−7} × 10000 × 5
= 20π × 10^{−3} T
Flux linking with per turn of the second solenoid = Bπr^{2} = Bπ × 10^{−4}
Total flux linking the second coil, Ï•_{1} = Bn_{2}πr^{2}
∴ Ï•_{1} = 100 × π × 10^{−4} × 20π × 10^{−3}
When the direction of the current is reversed, the total flux linking the second coil is given by
Ï•_{2} = −Bn_{2}πr^{2}
= −(100 × π × 10^{−4} × 20π × 10^{−3} )
The change in the flux through the second coil is given by
ΔÏ• = Ï•_{2} − Ï•_{1}
= 2 × (100 × π × 10^{−4} × 20π × 10^{−3})
Now,
e=∆ϕ∆t=4π2×104∆tThe current through the solenoid is given by
I=eR=4π2×104∆t×20The charge flown through the galvanometer is given by
q=I∆t=4π2×10420×dt×∆t =2×104 C
Question 13:
Figure shows a metallic square frame of edge a in a vertical plane. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the figure. Two boys pull the opposite corners of the square to deform it into a rhombus. They start pulling the corners at t = 0 and displace the corners at a uniform speed u. (a) Find the induced emf in the frame at the instant when the angles at these corners reduce to 60°. (b) Find the induced current in the frame at this instant if the total resistance of the frame is R. (c) Find the total charge which flows through a side of the frame by the time the square is deformed into a straight line.
Figure
Answer:
(a) The effective length of each side is the length that is perpendicular to the velocity of the corners.
Thus, the effective length of each side is a sin θ.
Net effective length for four sides = 4 ×
a2= 2a
∴ Induced emf = Bvl = 2Bau
(b) Current in the frame is given by
i
=eR=2auBR(c) Total charge q, which flows through the sides of the frame, is given by
q=∆ϕRHere,
ΔΦ = Change in the flux
R = Resistance of the coil
∴q=∆ϕR
=B(a20)R =Ba2R
Page No 307:
Question 14:
The north pole of a magnet is brought down along the axis of a horizontal circular coil (figure). As a result, the flux through the coil changes from 0.35 weber to 0.85 weber in an interval of half a second. Find the average emf induced during this period. Is the induced current clockwise or anticlockwise as you look into the coil from the side of the magnet ?
Figure
Answer:
Given:
Initial flux, Ï•_{1} = 0.35 weber
Final flux Ï•_{2} = 0.85 weber
∴ ΔÏ• = Ï•_{2} − Ï•_{1}
= (0.85 − 0.35) weber
= 0.5 weber
Also,
Δt = 0.5 s
The magnitude of the induced emf is given by
e=∆ϕ∆t=0.50.5=1 V
The induced current is anticlockwise when seen from the side of the magnet.
Question 15:
A wireloop confined in a plane is rotated in its own plane with some angular velocity. A uniform magnetic field exists in the region. Find the emf induced in the loop.
Answer:
When the wire loop is rotated in its own plane in a uniform magnetic field, the magnetic flux through it remains the same. Because there is no change in the magnetic flux, the emf induced in the wire loop is zero.
Question 16:
Figure shows a square loop of side 5 cm being moved towards right at a constant speed of 1 cm/s. The front edge enters the 20 cm wide magnetic field at t = 0. Find the emf induced in the loop at (a) t = 2 s, (b) t = 10 s, (c) t = 22 s and (d) t = 30 s.
Figure
Answer:
Given:
Initial velocity, u = 1 cm/s
Magnetic field, B = 0.6 T
(a) At t = 2 s:
Distance moved by the coil = 2 × 1 cm/s = 2 cm = 2 × 10
2 m
Area under the magnetic field at t = 2s, A = 2 × 5 × 10
4 m^{2}
Initial magnetic flux = 0
Final magnetic flux = BA = 0.6 × (10 × 10
4) Tm^{2}
Change in the magnetic flux, ΔÏ• = 0.6 × (10 × 10
4)
0
Now, induced emf in the coil is
e=∆ϕ∆t =0.6×(100)×1042 =3×104 V(b) At t = 10 s:
Distance moved by the coil = 10 × 1 = 10 cm
At this time square loop is completely inside the magnetic field, so there is no change in the flux linked with the coil with time.
Therefore, induced emf in the coil at this time is zero.
(c) At t = 22 s:
Distance moved = 22 × 1 = 22 cm
At this time loop is moving out of the field.
Initial magnetic flux = 0.6 × (5 × 5 × 10
4) Tm
At this time 2 cm part of the loop is ou t of the field.
Therefore, final magnetic flux = 0.6 × (3 × 5 × 10
4) Tm
Change in the magnetic flux, ΔÏ• = 0.6 × (3 × 5 × 10
4)
0.6 × (5 × 5 × 10
4) =
6 × 10
4 Tm^{2}
Now, induced emf is
e=∆ϕ∆t =6×1042 =3×104 V(d) At t = 30 s:
At this time loop is completely out of the field, so there is no change in the flux linked with the coil with time.
Therefore, induced emf in the coil at this time is zero.
Question 17:
Find the total heat produced in the loop of the previous problem during the interval 0 to 30 s if the resistance of the loop is 4.5 mâ„¦.
Answer:
Resistance of the coil, R = 45 mâ„¦ = 4.5 × 10^{−3} â„¦
The heat produced is found by taking the sum of the individual heats produced.
Thus, the net heat produced is given by
H = H_{1} + H_{2} + H_{3} + H_{4}
(a) Heat developed for the first 5 seconds:
Emf induced, e = 3 × 10^{−4} V
Current in the coil,
i=eR=3×1044.5×103= 6.7 × 10^{−2} A
H_{1} = (6.7 × 10^{−2})^{2} × 4.5 × 10^{−3} × 5
There is no change in the emf from 5 s to 20 s and from 25 s to 30 s.
Thus, the heat developed for the above mentioned intervals is given by
H_{2} = H_{4} = 0
Heat developed in interval t = 25 s to 30 s:
The current and voltage induced in the coil will be the same as that for the first 5 seconds.
H_{3} = (6.7 × 10^{−2})^{2} × 4.5 × 10^{−3} × 5
Total heat produced:
H = H_{1} + H_{3}
= 2 × (6.7 × 10^{−2})^{2} × 4.5 × 10^{−2} × 5
= 2 × 10^{−4} J
Question 18:
A uniform magnetic field B exists in a cylindrical region of radius 10 cm as shown in figure. A uniform wire of length 80 cm and resistance 4.0 â„¦ is bent into a square frame and is placed with one side along a diameter of the cylindrical region. If the magnetic field increases at a constant rate of 0.010 T/s, find the current induced in the frame.
Figure
Answer:
The magnetic field lines pass through coil abcd only in the part above the cylindrical region.
Radius of the cylindrical region, r = 10 cm
Resistance of the coil, R = 4 â„¦
The rate of change of the magnetic field in the cylindrical region is constant and is given by
dBdt=0.010 T/sThe change in the magnetic flux is given by
dϕdt=dBdtAThe induced emf is given by
e=dϕdt=dBdt×A=0.01π×r22 =0.01×3.14×0.012 =3.142×104=1.57×104 VThe current in the coil is given by
i=eR=1.57×1044 =0.39×104 =3.9×105 A
Question 19:
The magnetic field in the cylindrical region shown in figure increases at a constant rate of 20.0 mT/s. Each side of the square loop abcd and defa has a length of 1.00 cm and a resistance of 4.00 â„¦. Find the current (magnitude and sense) in the wire ad if (a) the switch S_{1} is closed but S_{2} is open, (b) S_{1} is open but S_{2} is closed, (c) both S_{1} and S_{2} are open and (d) both S_{1} and S_{2} are closed.
Figure
Answer:
(a) When switch S_{1} is closed and switch S_{2} is open:
Rate of change of the magnetic field = 20 mT/s =
2×104 T/sNet resistance of the coil adef, R = 4 × 4 = 16 â„¦
Area of the coil adef = (10^{−2} )^{−2} = 10^{−4} m^{2}
The emf induced is given by
e=dϕdt=A.dBdt = 10^{−4} × 2 × 10^{−2}
= 2 × 10^{−6} V
The current through the wire ad is given by
i
=eR=2×10616 = 1.25 × 10^{−7} A along ad
(b) When switch S_{2} is closed and switch S_{1} is open:
Net resistance of the coil abcd, R = 16 â„¦
The induced emf is given by
e=A×dBdt=2×106 VThe current through wire ad is given by
i=20×10616=1.25×107 Aalong da
(c) When both S_{1} and S_{2} are open, no current is passed, as the circuit is open. Thus, i = 0.
(d) When both S_{1} and S_{2} are closed, the circuit forms a balanced a Wheatstone bridge and no current flows along ad. Thus, i = 0.
Question 20:
Figure shows a circular coil of N turns and radius a, connected to a battery of emf ε through a rheostat. The rheostat has a total length L and resistance R. the resistance of the coil is r. A small circular loop of radius a‘ and resistance r‘ is placed coaxially with the coil. The centre of the loop is at a distance x from the centre of the coil. In the beginning, the sliding contact of the rheostat is at the left end and then onwards it is moved towards right at a constant speed v. Find the emf induced in the small circular loop at the instant (a) the contact begins to slide and (b) it has slid through half the length of the rheostat.
Figure
Answer:
The magnetic field due to coil 1 at the centre of coil 2 is given by
B=μ0 Nia22 (a2+x2)3/2
The flux linked with coil 2 is given by
ϕ=B.A’=μ0 Nia22 (a2+x2)3/2πa’2Now, let y be the distance of the sliding contact from its left end.
Given:
v=dydtTotal resistance of the rheostat = R
When the distance of the sliding contact from the left end is y, the resistance of the rheostat (R‘) is given by
R’=RLyThe current in the coil is the function of distance y travelled by the sliding contact of the rheostat. It is given by
i=εRLy+rThe magnitude of the emf induced can be calculated as:
e=dϕdt=μ0 Na2 a’2π 2 (a2+x2)3/2didt ⇒e=μ0 N πa2 a’22 (a2+x2)3/2ddtεRLy+r⇒e=μ0 N πa2 a’22 (a2+x2)3/2εRLvRLy+r2(a) For y = L,
e=μ0N πa2 a’2ε Rv2L (a2+x2)3/2 (R+r)2(b) For y = L/2,
RLy=R2⇒e=μ0N πa2 a’22L(a2+x2)3/2εRvR2+r2
Question 21:
A circular coil of radius 2.00 cm has 50 turns. A uniform magnetic field B = 0.200 T exists in the space in a direction parallel to the axis of the loop. The coil is now rotated about a diameter through an angle of 60.0°. The operation takes 0.100 s. (a) Find the average emf induced in the coil. (b) If the coil is a closed one (with the two ends joined together) and has a resistance of 4.00 â„¦, calculate the net charge crossing a crosssection of the wire of the coil.
Answer:
Given:
Number of turns of the coil, N = 50
Magnetic field through the circular coil,
B→= 0.200 T
Radius of the circular coil, r = 2.00 cm = 0.02 m
Angle through which the coil is rotated, θ = 60°
Time taken to rotate the coil, t = 0.100 s
(a) The emf induced in the coil is given by
e=N∆ϕ∆t=N(Bf→.A→fBi→.A→i)T =NB.A (cos0ocos 60°)T =50×2×101×π(0.02)22×0.1 =5×4×105×π =2π×102 V=6.28×103 V(b) The current in the coil is given by
i=eR=6.28×1034 =1.57×103 AThe net charge passing through the cross section of the wire is given by
Q=it=1.57×103×101 =1.57×104 C
Question 22:
A closed coil having 100 turns is rotated in a uniform magnetic field B = 4.0 × 10^{−4} T about a diameter which is perpendicular to the field. The angular velocity of rotation is 300 revolutions per minute. The area of the coil is 25 cm^{2} and its resistance is 4.0 â„¦. Find (a) the average emf developed in half a turn from a position where the coil is perpendicular to the magnetic field, (b) the average emf in a full turn and (c) the net charge displaced in part (a).
Answer:
Given:
Number of turns in the coil, n = 100 turns
Magnetic field, B = 4 × 10^{−4}
Area of the loop, A = 25 cm^{2} = 25 × 10^{−4} m^{2}
(a) When the coil is perpendicular to the field:
Ï•_{1} = nBA
When the coil goes through the half turn:
Ï•_{2} = nBA cos 180° = −nBA
∴ ΔÏ• = 2nBA
When the coil undergoes 300 revolutions in 1 minute, the angle swept by the coil is
300 × 2π rad/min = 10π rad/s
10π rad is swept in 1 s.
π rad is swept in
110ππ=110 s
e=dϕdt=2nBAdt =2×100×4×104×25×1041/10 =2×103 V(b) Ï•_{1} = nBA, Ï•_{2} = nBA (θ = 360°)
ΔÏ• = 0, thus emf induced will be zero.
(c) The current flowing in the coil is given by
i=eR=2×1034=12×103 = 0.5 × 10^{−3} = 5 × 10^{−4} A
Hence, the net charge is given by
Q = idt = 5 × 10^{−4} ×
110 = 5 × 10^{−5} C
Question 23:
A coil of radius 10 cm and resistance 40 â„¦ has 1000 turns. It is placed with its plane vertical and its axis parallel to the magnetic meridian. The coil is connected to a galvanometer and is rotated about the vertical diameter through an angle of 180°. Find the charge which flows through the galvanometer if the horizontal component of the earth’s magnetic field is B_{H} = 3.0 × 10^{−5} T.
Answer:
Given:
Radius of the coil, r = 10 cm = 0.1 m
Resistance of the coil, R = 40 â„¦
Number of turns in the coil, N = 1000
Angle of rotation, θ = 180°
Horizontal component of Earth’s magnetic field, B_{H} = 3 × 10^{−5} T
Magnetic flux, Ï• = NBA cos 180°
⇒ Ï• = −NBA
= −1000 × 3 × 10^{−5} × π × 1 × 1 × 10^{−2}
= 3π × 10^{−4} Wb
dÏ• = 2NBA = 6π × 10^{−4} Wb
e=dϕdt=6π×104 dtV
Thus, the current flowing in the coil and the total charge are:
i=eR=6π×10440dt=4.71×105dtQ=4.71×105×dtdt =4.71×105 C
Question 24:
A circular coil of one turn of radius 5.0 cm is rotated about a diameter with a constant angular speed of 80 revolutions per minute. A uniform magnetic field B = 0.010 T exists in a direction perpendicular to the axis of rotation. Find (a) the maximum emf induced, (b) the average emf induced in the coil over a long period and (c) the average of the squares of emf induced over a long period.
Answer:
Given,
Radius of the circular coil, R = 5.0 cm
Angular speed of circular coil,
ω= 80 revolutions/minute
Magnetic field acting perpendicular to the axis of rotation, B = 0.010 T
The emf induced in the coil
eis given by,
e=dϕdt⇒e=dB.Acosθdt⇒e=BAsinθdθdt
⇒e = −BAωsinθ
(
dθdt=ω= the rate of change of angle between the arc vector and B)
(a) For maximum emf, sinθ = 1
∴e = BAω
⇒e = 0.010 × 25 × 10^{−4} × 80 ×
2π×π60
⇒e = 0.66 × 10^{−3} = 6.66 × 10^{−4} V
(b) The direction of the induced emf changes every instant. Thus, the average emf becomes zero.
(c) The emf induced in the coil is e = −BAωsinθ = −BAωsin ωt
The average of the squares of emf induced is given by
eav2=∫0TB2A2ω2sin2ωt dt∫0Tdt⇒eav2=B2A2ω2∫0Tsin2ωt dt∫0Tdt⇒eav2=B2A2ω2∫0T1cos2ωt dt2T⇒eav2=B2A2ω22Ttsin2ωt2ω0T⇒eav2=B2A2ω22TTsin4πsin 02ω=B2A2ω22⇒eav2=(6.66×104)22=22.1778×108 V2 ∵BAω=6.66×104 V⇒eav2 =2.2×107 V2
Page No 308:
Question 25:
Suppose the ends of the coil in the previous problem are connected to a resistance of 100 â„¦. Neglecting the resistance of the coil, find the heat produced in the circuit in one minute.
Answer:
Given:
T = 1 minute
Heat produced in the circuit is calculated using the following relation:
H=∫0Ti2Rdt
⇒H=∫01 minB2A2ω2R2sinωtRdt
=B2A2ω22R. ∫01 min1cos 2ωt dt=B2A2ω22R1sin2ωt2ω01 min=B2A2ω22R60sin 2×80×2π/60×602×80×2π/60=602R×π2r4×B2×80×2π602=60200×10×649×10×625×108×104=625×6×649×2×1011=1.33×107 J
Question 26:
Figure
Figure shows a circular wheel of radius 10.0 cm whose upper half, shown dark in the figure, is made of iron and the lower half of wood. The two junctions are joined by an iron rod. A uniform magnetic field B of magnitude 2.00 × 10^{−4} T exists in the space above the central line as suggested by the figure. The wheel is set into pure rolling on the horizontal surface. If it takes 2.00 seconds for the iron part to come down and the wooden part to go up, find the average emf induced during this period.
Answer:
Magnetic flux through the wheel (initially):
ϕ1=BA=2×104×π0.122 =π×106 WbAs the wheel rotates, the wooden (nonmetal) part of the wheel comes inside the magnetic field and the iron part of the wheel comes outside the magnetic field. Thus, the magnetic flux through the wheel becomes zero.
i.e.
ϕ2=0dt = 2 s
The average emf induced in the wheel is given by
e=dϕdt =ϕ2ϕ1dt =π×1062 =1.57×106 V
Question 27:
A 20 cm long conducting rod is set into pure translation with a uniform velocity of 10 cm s^{−1} perpendicular to its length. A uniform magnetic field of magnitude 0.10 T exists in a direction perpendicular to the plane of motion. (a) Find the average magnetic force on the free electrons of the rod. (b) For what electric field inside the rod, the electric force on a free elctron will balance the magnetic force? How is this electric field created? (c) Find the motional emf between the ends of the rod.
Answer:
Given:
Length of the rod, l = 20 cm = 0.2 m
Velocity of the rod, v = 10 cm/s = 0.1 m/s
Magnetic field, B = 0.10 T
(a) The force on a charged particle moving with velocity v in a magnetic field is given by
F→=qv→×B→F = qvB sin θ
Here,
θ = 90^{o}
Now,
F = (1.6 × 10^{−19}) × (1 × 10^{−1}) × (1 × 10^{−1})
= 1.6 × 10^{−21} N
(b) The electrostatic force on the charged particle is qE.
Here,
qE = qvB
⇒ E = (1 × 10^{−1} ) × (1 × 10^{−1})
= 1 × 10^{−2} V/m
It is created because of the induced emf.
(c) Motional emf between the ends of the rod, e = Bvl
⇒ e = 0.1 × 0.1 × 0.2
= 2 × 10^{−3} V
Question 28:
A metallic metre stick moves with a velocity of 2 m s^{−1} in a direction perpendicular to its length and perpendicular to a uniform magnetic field of magnitude 0.2 T. Find the emf induced between the ends of the stick.
Answer:
Given:
Length of the stick, l = 1 m
Magnetic field, B = 0.2 T
Velocity of the stick, v = 2 m/s
Thus, we get
Induced emf, e = Blv = 0.2 × 1 × 2 = 0.4 V
Question 29:
A 10 m wide spacecraft moves through the interstellar space at a speed 3 × 10^{7} m s^{−1}. A magnetic field B = 3 × 10^{−10} T exists in the space in a direction perpendicular to the plane of motion. Treating the spacecraft as a conductor, calculate the emf induced across its width.
Answer:
Given:
l = 10 m
v = 3 × 10^{7} m/s
B = 3 × 10^{−10} T
Now,
Motional emf = Bvl
= (3 × 10^{−10} ) × (3 × 10^{7} ) × (10)
= 9 × 10^{−2}
= 0.09 V
Question 30:
The two rails of a railway track, insulated from each other and from the ground, are connected to a millivoltmeter. What will be the reading of the millivoltmeter when a train travels on the track at a speed of 180 km h^{−1}? The vertical component of earth’s magnetic field is 0.2 × 10^{−4} T and the rails are separated by 1 m.
Answer:
Here,
Velocity of the train, v = 180 km/h = 50 m/s
Earth’s magnetic field, B = 0.2 × 10^{−4} T
Separation between the railings, l = 1 m
Induced emf, e = Bvl = 0.2 × 10^{−4} × 50
= 10^{−3} V
So, the voltmeter will record 1 mV as the reading.
Question 31:
A rightangled triangle abc, made from a metallic wire, moves at a uniform speed v in its plane as shown in figure. A uniform magnetic field B exists in the perpendicular direction. Find the emf induced (a) in the loop abc, (b) in the segment bc, (c) in the segment ac and (d) in the segment ab.
Figure
Answer:
(a) The emf induced in loop abc is zero, as there is no change in the magnetic flux through it.
(b) The emf induced is given by
e=v→×B→.l→
Emf induced in segment bc, e = Bvl (With positive polarity at point C)
(c) There is no emf induced in segment bc, as the velocity is parallel to its length.
(d) The emf induced in segment ab is calculated by the following formula:
e = B.v. (Effective length of ab)
The effective length of ab is along the direction perpendicular to its velocity.
Emf induced, e = B.v.(bc)
Question 32:
A copper wire bent in the shape of a semicircle of radius r translates in its plane with a constant velocity v. A uniform magnetic field B exists in the direction perpendicular to the plane of the wire. Find the emf induced between the ends of the wire if (a) the velocity is perpendicular to the diameter joining free ends, (b) the velocity is parallel to this diameter.
Answer:
(a) The emf induced between the ends of the wire is calculated using the following formula:
e = Bv (Effective length of the wire)
Effective length of the wire = Component of length perpendicular to the velocity
Here, the component of length moving perpendicular to v is 2r.
∴ Induced emf, e = Bv2r
(b) When the velocity is parallel to the diameter of the semicircular wire, the component of its length perpendicular to its velocity is zero.
∴ Induced emf, e = 0
Question 33:
A wire of length 10 cm translates in a direction making an angle of 60° with its length. The plane of motion is perpendicular to a uniform magnetic field of 1.0 T that exists in the space. Find the emf induced between the ends of the rod if the speed of translation is 20 cm s^{−1}.
Answer:
Given:
Length of the rod, l = 10 cm = 0.1 m
Angle between the velocity and length of the rod, θ = 60°
Magnetic field, B = 1 T
Velocity of the rod, v = 20 cm/s = 0.2 m/s
The motional emf induced in the rod is given by
e=v→×B→.l→∴ e = Bvl sin 60°
We take the component of the length vector that is perpendicular to the velocity vector.
∴ e =
1×0.2×0.1×32 = 17.32 × 10^{−3} V
Question 34:
A circular copperring of radius r translates in its plane with a constant velocity v. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the ring. Consider different pairs of diametrically opposite points on the ring. (a) Between which pair of points is the emf maximum? What is the value of this maximum emf? (b) Between which pair of points is the emf minimum? What is the value of this minimum emf ?
Answer:
(a) The maximum value of the emf is between the end points of the diameter perpendicular to the velocity.
The value of the maximum emf is given by
E_{max} = vB(2R)
(b) The minimum value of emf is between the end points of the diameter parallel to the velocity of the ring.
The minimum value of emf is given by
E_{min} = 0
Question 35:
Figure shows a wire sliding on two parallel, conducting rails placed at a separation l. A magnetic field B exists in a direction perpendicular to the plane of the rails. What force is necessary to keep the wire moving at a constant velocity v ?
Figure
Answer:
Because the force exerted by the magnetic field on the rod is given by F_{magnetic} = ilB, the direction of this force is opposite to that of the motion of the rod.
Now, let the external force on it be F.
Because the velocity is constant, the net force acting on the wire must be zero.
Thus, F = F_{magnetic} = ilB is acting in the direction of the velocity.
Question 36:
Figure shows a long Ushaped wire of width l placed in a perpendicular magnetic field B. A wire of length l is slid on the Ushaped wire with a constant velocity v towards right. The resistance of all the wires is r per unit length. At t = 0, the sliding wire is close to the left edge of the Ushaped wire. Draw an equivalent circuit diagram, showing the induced emf as a battery. Calculate the current in the circuit.
Figure
Answer:
The induced emf is given by
e = Bvl
Total resistance, R = r × Total length of the wire
Because the length of the movable wire is l and the distance travelled by the movable wire in time t is vt, the total length of the loop is 2 (l + vt).
∴ e = i × 2r (l + vt)
Bvl = 2ri (l + vt)
⇒i=Bvl2r(l+vt)
Question 37:
Consider the situation of the previous problem. (a) Calculate the force needed to keep the sliding wire moving with a constant velocity v. (b) If the force needed just after t = 0 is F_{0}, find the time at which the force needed will be F_{0}/2.
Answer:
Emf induced in the circuit, e = Bvl
Current in the circuit,
i=eR=Bvl2r(l+vt)(a) Force F needed to keep the sliding wire moving with a constant velocity v will be equal in magnitude to the magnetic force on it. The direction of force F will be along the direction of motion of the sliding wire.
Thus, the magnitude of force F is given by
F=ilB=Bvl2r(l+vt)×lB =B2l2v2r(l+vt)(b) The magnitude of force F at t = 0 is given by
F0=ilB=lBlBv2rl =lB2v2r …(1)Let at time t = T, the value of the force be F_{0}/2.
Now,
F02=l2B2v2r(l+vT)On substituting the value of F_{0} from (1), we get
lB2v4r=l2B2v2r(l+vT)⇒2l=l+vT⇒T=lv
Page No 309:
Question 38:
Consider the situation shown in figure. The wire PQ has mass m, resistance r and can slide on the smooth, horizontal parallel rails separated by a distance l. The resistance of the rails is negligible. A uniform magnetic field B exists in the rectangular region and a resistance R connects the rails outside the field region. At t = 0, the wire PQ is pushed towards right with a speed v_{0}. Find (a) the current in the loop at an instant when the speed of the wire PQ is v, (b) the acceleration of the wire at this instant, (c) the velocity v as a functions of x and (d) the maximum distance the wire will move.
Figure
Answer:
(a) When wire PQ is moving with a speed v, the emf induced across it is given by
e = Blv
Total resistance of the circuit = r + R
∴ Current in the circuit, i =
Blvr+R(b) Force acting on the wire at the given instant, F = ilB
On substituting the value of i from above, we get
F=(Blv)(lB)(R+r)=B2l2vR+rAcceleration of the wire is given by
a
=B2l2vm (R+r)(c) Velocity can be expressed as:
v = v_{0} + at =
v0B2l2vm (R+r)t (As force is opposite to velocity)
Velocity as the function of x is given by
v=v0B2l2xm (R+r) d a=vdvdx=B2l2vm (R+r)dx= m (R+r)B2l2dvOn integrating both sides, we get
x=m(R+r)v0B2l2
Question 39:
A rectangular frame of wire abcd has dimensions 32 cm × 8.0 cm and a total resistance of 2.0 â„¦. It is pulled out of a magnetic field B = 0.020 T by applying a force of 3.2 × 10^{−5} N (figure). It is found that the frame moves with constant speed. Find (a) this constant speed, (b) the emf induced in the loop, (c) the potential difference between the points a and b and (d) the potential difference between the points c and d.
Figure
Answer:
Given:
Total resistance of the frame, R = 2.0 â„¦
Magnetic field, B = 0.020 T
Dimensions of the frame:
Length, l = 32 cm = 0.32 m
Breadth, b = 8 cm = 0.08 m
(a) Let the velocity of the frame be v.
The emf induced in the rectangular frame is given by
e = Blv
Current in the coil,
i=BlvRThe magnetic force on the rectangular frame is given by
F = ilB = 3.2 × 10^{−5} N
On putting the value of i, we get
B2l2vR=3.2×105⇒(0.020)2×(0.08)2×v2=3.2×105⇒v=3.2×1056.4×103×4×104 =25 m/s(b) Emf induced in the loop, e = vBl
⇒ e = 25 × 0.02 × 0.08
= 4 × 10^{−2} V
(c) Resistance per unit length is given by
r
=20.8Ratio of the resistance of part,
adcb=2×0.720.8=1.8 Ω
Vab=iR=Blv2×1.8 =0.2×0.08×25×1.82 =0.036 V=3.6×102 V(d) Resistance of cd:
Rcd=2×0.80.8=0.2 ΩV=iRcd=2×0.08×25×0.22 =4×103 V
Question 40:
Figure shows a metallic wire of resistance 0.20 â„¦ sliding on a horizontal, Ushaped metallic rail. The separation between the parallel arms is 20 cm. An electric current of 2.0 µA passes through the wire when it is slid at a rate of 20 cm s^{−1}. If the horizontal component of the earth’s magnetic field is 3.0 × 10^{−5} T, calculate the dip at the place.
Figure
Answer:
Given:
Separation between the parallel arms, l = 20 cm = 20 × 10^{−2} m
Velocity of the sliding wire, v = 20 cm/s = 20 × 10^{−2} m/s
Horizontal component of the earth’s magnetic field, B_{H} = 3 × 10^{−5} T
Current through the wire, i = 2 µA = 2 × 10^{−6} A
Resistance of the wire, R = 0.2 â„¦
Let the vertical component of the earth’s magnetic field be B_{v} and the angle of the dip be δ.
Now,
i=Bv lvR ⇒Bv=iRlv
=2×105×2×10120×102×20×102=2×2×1072×2×102 =1×105 T We know,
tanδ=BvBH=1×1053×105=13⇒δ=tan113
Question 41:
A wire ab of length l, mass m and resistance R slides on a smooth, thick pair of metallic rails joined at the bottom as shown in figure. The plane of the rails makes an angle θ with the horizontal. A vertical magnetic field B exists in the region. If the wire slides on the rails at a constant speed v, show that
B=mg R sinθvl2cos2 θFigure
Answer:
Component of weight along its motion, F‘ = mgsinθ
The emf induced in the rod due to its motion is given by
e = Bl’v’
Here,
l‘ = Component of the length of the rod perpendicular to the magnetic field
v‘ = Component of the velocity of the rod perpendicular to the magnetic field
i=B×l×v cosθR
F→=il→×B→=ilBsin(90θ)F=ilB=Blv cosθR×l×BcosθF=B2l2vcos2θRThe direction of force F is opposite to F.‘
Because the rod is moving with a constant velocity, the net force on it is zero.
Thus,
F
F’ = 0
F = F‘
or
B2l2v cos2 θR=mgsinθ
∴B=Rmgsinθl2vcos2θ
Question 42:
Consider the situation shown in figure. The wires P_{1}Q_{1} and P_{2}Q_{2} are made to slide on the rails with the same speed 5 cm s^{−1}. Find the electric current in the 19 â„¦ resistor if (a) both the wires move towards right and (b) if P_{1}Q_{1} moves towards left but P_{2}Q_{2} moves towards right.
Figure
Answer:
(a) When both wires move in same direction:
The sliding wires constitute two parallel sources of emf.
The net emf is given by
e = Blv
⇒ e = (1 × 4 × 10^{−2} ) × 5 × (10^{−2})
= 20 × 10^{−4} V
The resistance of the sliding wires is 2 â„¦.
∴ Net resistance =
2×22+2+ 19 = 20 â„¦
Net current through 19 â„¦ =
2×10420= 0.1 mA
(b) When both wires move in opposite directions with the same speed, the direction of the emf induced in both of them is opposite. Thus, the net emf is zero.
∴ Net current through 19 â„¦ = 0
Question 43:
Suppose the 19 â„¦ resistor of the previous problem is disconnected. Find the current through P_{2}Q_{2} in the two situations (a) and (b) of that problem.
Answer:
(a) When the wires move in the same direction, their polarity remains the same. The circuit remains incomplete. Therefore, no current flows in the circuit.
(b) When the wires move in opposite directions, their polarities are reversed. Thus, current flows in the circuit.
VP2Q2=Blv = 1 × 0.04 × 0.05
= 2 × 10^{−3} V
R = 2 â„¦
Current in the circuit is given by
i=2×1032 = 1 × 10^{−3} A = 1 mA
Question 44:
Consider the situation shown in figure. The wire PQ has a negligible resistance and is made to slide on the three rails with a constant speed of 5 cm s^{−1}. Find the current in the 10 â„¦ resistor when the switch S is thrown to (a) the middle rail (b) the bottom rail.
Figure
Answer:
Given:
Magnetic field, B = 1 T
Velocity of the sliding wire, v = 5 × 10^{−2} m/s
Resistance of the connected resistor, R = 10 â„¦
(a) When the switch is thrown to the middle rail:
Length of the sliding wire = 2 × 10^{−2} m
Induced emf, E = Bvl
= 1 × (5 × 10^{−2}) × (2 × 10^{−2}) V
= 10 × 10^{−4} = 10^{−3} V
Current in the 10 â„¦ resistor is given by
i=ER =10310=104=0.1 mA(b) When the switch is thrown to the bottom rail:
The length of the sliding wire becomes 4 × 10^{−2} m.
The induced emf is given by
E = Bvl‘
= 1 × (5 × 10^{−2}) × (4 × 10^{−2})
= 20 × 10^{−4} V
Now,
Current, i =
20×10410A
= 2 × 10^{−4} A = 0.2 mA
Question 45:
The current generator I_{g}‘ shown in figure, sends a constant current i through the circuit. The wire cd is fixed and ab is made to slide on the smooth, thick rails with a constant velocity v towards right. Each of these wires has resistance r. Find the current through the wire cd.
Figure
Answer:
Current passing through the circuit initially = i
Initial emf = ir
Emf induced due to motion of ab, e = Blv
Net emf, e_{net}= ir − Blv
Net resistance = 2r
Thus, the current passing through the circuit is
irBlv2r.
Question 46:
The current generator I_{g}‘ shown in figure, sends a constant current i through the circuit. The wire ab has a length l and mass m and can slide on the smooth, horizontal rails connected to I_{g}. The entire system lies in a vertical magnetic field B. Find the velocity of the wire as a function of time.
Figure
Answer:
Because current i passes through the sliding wire, the magnetic force on the wire (F) is ilB.
Now,
Acceleration of the sliding wire, a =
ilBmVelocity of the sliding wire, v = u + at
âˆµ u = 0
∴ v =
ilBtm
Question 47:
The system containing the rails and the wire of the previous problem is kept vertically in a uniform horizontal magnetic field B that is perpendicular to the plane of the rails (figure). It is found that the wire stays in equilibrium. If the wire ab is replaced by another wire of double its mass, how long will it take in falling through a distance equal to its length?
Figure
Answer:
Let us consider the above free body diagram.
As the net force on the wire is zero, ilB = mg.
When the wire is replaced by a wire of double mass, we have
Now, let a’ be the acceleration of the wire in downward direction and t be the time taken by the wire to fall.
Net force on the wire = 2mg − ilB = F_{net}
On applying Newton’s second law, we get
2mg − ilB = 2 ma‘ …(1)
⇒a’=2mgilB2 ms=ut+12a’t2⇒l=12×2mgilB2m×t2 [∵s=l]⇒t=4 ml2mgilB⇒t=4 ml2mgmg [From (1)]⇒t=2lg
Page No 310:
Question 48:
The rectangular wireframe, shown in figure, has a width d, mass m, resistance R and a large length. A uniform magnetic field B exists to the left of the frame. A constant force F starts pushing the frame into the magnetic field at t = 0. (a) Find the acceleration of the frame when its speed has increased to v. (b) Show that after some time the frame will move with a constant velocity till the whole frame enters into the magnetic field. Find this velocity v_{0}. (c) Show that the velocity at time t is given by
v = v_{0}(1 − e^{−Ft}^{/mv0}).
Figure
Answer:
Given:
Width of rectangular frame = d
Mass of rectangular frame = m
Resistance of the coil = R
(a) As the frame attains the speed v
Emf developed in side AB = Bdv (When it attains a speed v)
Current =
BdvRThe magnitude of the force on the current carrying conductor moving with speed v in direction perpendicular to the magnetic field as well as to its length is given by
F=ilBTherefore, Force F_{B} =
Bd2vRAs the force is in direction opposite to that of the motion of the frame .
Therefore,Net force is given by
Fnet = FFBFnet=FBd2v2 R=RFBd2vRApplying Newton’s second law
RFBd2v2R=maNet acceleration is given by a=
RFBd2vmR(b)
Velocity of the frame becomes constant when its acceleration becomes 0.
Let the velocity of the frame be v_{0}
FmB2d2v0mR= 0⇒Fm=B2d2v0mR⇒v0=FRB2d2As the speed thus calculated depends on F, R, B and d all of them are constant, therefore the velocity is also constant.
Hence, proved that the frame moves with a constant velocity till the whole frame enters.
(c)
Let the velocity at time t be v.
The acceleration is given by
a=dvdt⇒RFd2B2v2mR=dvdtdvRFd2B2v2 =dtmRIntegrating⇒∫0vdvRFd2B2v2=∫0tdtmR⇒ln(RFd2B2v)0v=d2B2tRm0t⇒ln(RFd2B2v)ln (RF) = d2B2tRm⇒d2B2vRF=1ed2B2tRmv=FRl2B21eB2d2v0tRv0mv=v0(1eFt/v0m) ∵ F=B2d2v0R
Question 49:
Figure shows a smooth pair of thick metallic rails connected across a battery of emf ε having a negligible internal resistance. A wire ab of length l and resistance r can slide smoothly on the rails. The entire system lies in a horizontal plane and is immersed in a uniform vertical magnetic field B. At an instant t, the wire is given a small velocity v towards right. (a) Find the current in it at this instant. What is the direction of the current? (b) What is the force acting on the wire at this instant? (c) Show that after some time the wire ab will slide with a constant velocity. Find this velocity.
Figure
Answer:
According to Fleming’s left hand rule the force in the wire ab will be in the upward direction.
Moreover, a moving wire ab is equivalent to a battery of emf vBl as shown in the figure.
At the given instant, the net emf across the wire (e) is E − Bvl.
(a) The current through the wire is given by
i=EBvlrThe direction of the current is from b to a.
(b) The force acting on the wire at the given instant is given by
F=ilB=EBvlrtowards right
(c) The velocity of the wire attains a value such that it satisfies E = Bvl.
The net force on the wire becomes zero. Thus, the wire moves with a constant velocity v.
∴
v=EBl
Question 50:
A conducting wire ab of length l, resistance r and mass m starts sliding at t = 0 down a smooth, vertical, thick pair of connected rails as shown in figure. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the rails. (a) Write the induced emf in the loop at an instant t when the speed of the wire is v. (b) What would be the magnitude and direction of the induced current in the wire? (c) Find the downward acceleration of the wire at this instant. (d) After sufficient time, the wire starts moving with a constant velocity. Find this velocity v_{m}. (e) Find the velocity of the wire as a function of time. (f) Find the displacement of the wire as a function of time. (g) Show that the rate of heat developed in the wire is equal to the rate at which the gravitational potential energy is decreased after steady state is reached.
Figure
Answer:
(a) When the speed of the wire is v, the emf developed in the loop is, e = Blv.
(b)
Magnitude of the induced current in the wire, I =
BlvRAs wire is moving the magnetic flux passing through the loop is increasing with time. Therefore, the direction of the current should be as such to oppose the change in magnetic flux. Therefore in order to induce the current in anticlockwise direction the current flows from b to a
(c)
Due to motion of the wire in the magnetic field there is a force in upward direction (perpendicular to the wire).
The magnitude of the force on the wire carrying current i is given by
F=ilBThe net force on the wire =
mgF=mgilBDownward acceleration of the wire due to current, a
=mgFm
a=mgilBma=gB2l2vRm ∵ i=BlvR(d) Let the wire start moving with a constant velocity.
Now,
Let the speed of the wire be v_{m}
As speed is constant, acceleration, a = 0
a=gB2l2vmRm =0
B2l2vmRm=g⇒vm=gRmB2l2(e)
The acceleration of the wire can be expressed as time rate of change of velocity
dvdt=a
∴dvdt=mgB2l2v/Rm⇒dvmgB2l2v/Rm=dt∫0vm dvmgB2l2vR=∫dt0t⇒mB2l2RlogmgB2l2vR0v=t⇒mRB2l2logmgB2l2vRlog(mg)=t⇒logmgB2l2vRmg=tB2l2mR⇒log1B2l2vRmg=tB2l2mR⇒1B2l2vRmg=etB2l2mR⇒1etB2l2mR=B2l2vRmgv=RmgB2l21eB2l2tmR⇒v=vm1egtvm ∵vm=RmgB2l2(f)
The velocity can be expressed as time rate of change of position
x is the position of the wire at instant t.
âˆµ
dxdt=vThus, the displacement of the wire can be expressed as:
s=∫0tdx=∫0tvdt
∴s=xtx0=vm∫0t1egtvm.dts=vm.t+vmg.egtvm0ts=vmt+vm2gegtvmvm2gs=vmtvm2g1egtvm(g)
Rate of development of heat in the wire is given by P = V × i
V=Blvi=BlvRTherefore, the rate of development of heat in the wire is given by
P=Blv×BlvR=B2l2v2RP=B2l2vm21egtvm2R ∵v=vm1egtvmRate of decrease in potential energy is given by
dUdt=ddt(mgx)=mg.dxdtdUdt=mgvdUdt=mg.vm1egtvm ∵v=vm1egtvmAfter the steady state, i.e., t → ∞,
dUdt=mgvmP=l2B2Rvm2P=l2B2R×vm×mgRl2B2 ∵vm=mgRl2B2P=mgvmThus, after the steady state,
P=dUdt
Question 51:
A bicycle is resting on its stand in the eastwest direction and the rear wheel is rotated at an angular speed of 100 revolutions per minute. If the length of each spoke is 30.0 cm and the horizontal component of the earth’s magnetic field is 2.0 × 10^{−5} T, find the emf induced between the axis and the outer end of a spoke. Neglect centripetal force acting on the free electrons of the spoke.
Answer:
Given:
Length of the spoke of the bicycle’s wheel, l = 0.3 m
Earth’s magnetic field,
B→= 2.0 × 10^{−5} T
Length of each spoke = 30.0 cm = 0.3 m
Angular speed of the wheel,
ω=10060×2π=103π rad/sLinear speed of the spoke,
v=l2×ω=0.32×103πNow,
Emf induced in the spoke of the wheel, e = Blv
⇒e=(2.0×105)×(0.3)×0.32×103×π = 3
π× 10^{−6} V
= 3 × 3.14 × 10^{−6}V
= 9.42 × 10^{−6} V
Question 52:
A conducting disc of radius r rotates with a small but constant angular velocity ω about its axis. A uniform magnetic field B exists parallel to the axis of rotation. Find the motional emf between the centre and the periphery of the disc.
Answer:
The angular velocity of the disc is ω. Also, the magnetic field of magnitude B is perpendicular to the disc.
Let us take a circular element of thickness da at a distance a from the centre.
Linear speed of the element at a from the centre, v = ωa
Now,
de=Blvde=B×da×aω⇒e=∫0rBωada e=12Bωr2
Question 53:
Figure shows a conducting disc rotating about its axis in a perpendicular magnetic field B. A resistor of resistance R is connected between the centre and the rim. Calculate the current in the resistor. Does it enter the disc or leave it at the centre? The radius of the disc is 5.0 cm, angular speed ω = 10 rad/s, B = 0.40 T and R = 10 â„¦.
Figure
Answer:
Given:
Magnetic field perpendicular to the disc, B = 0.40 T
Angular speed, ω = 10 rad/s
Resistance, R = 10 â„¦
Radius of the disc, r = 5 cm = 0.5 m
Let us consider a rod of length 0.05 m fixed at the centre of the disc and rotating with the same ω.
Now,
v=l2×ω=0.052×10e=Blv =0.40×0.05×0.052×10 =5×103 Vi=eR =5×10310=0.5 mAAs the disc is rotating in the anticlockwise direction, the emf induced in the disc is such that the centre is at the higher potential and the periphery is at the lower potential. Thus, the current leaves from the centre.
Question 54:
The magnetic field in a region is given by
B→=k→B0Lywhere L is a fixed length. A conducting rod of length L lies along the Yaxis between the origin and the point (0, L, 0). If the rod moves with a velocity v = v_{0}
i→, find the emf induced between the ends of the rod.
Answer:
Magnetic field in the given region,
B→=B0Lyk^Length of the rod on the yaxis = L
Velocity of the rod, v = v_{0}
i^We will consider a small element of length dy on the rod.
Now,
Emf induced in the element:
de = Bvdy
⇒de=B0Ly×v0×dy =B0v0LydyAnd,e=B0v0L∫0Lydy =B0v0Ly220L =B0v0LL22 =12B0v0L
Question 55:
Figure shows a straight, long wire carrying a current i and a rod of length l coplanar with the wire and perpendicular to it. The rod moves with a constant velocity v in a direction parallel to the wire. The distance of the wire from the centre of the rod is x. Find the motional emf induced in the rod.
Figure
Answer:
Here, the magnetic field
B→due to the long wire varies along the length of the rod. We will consider a small element of the rod of length da at a distance a from the wire. The magnetic field at a distance a is given by
B→=μ0i2πaNow,
Induced emf in the rod:
de=Bvda =μ0i2πa×v×daIntegrating from
xl2to
x+l2, we get
e=∫xl2x+l2de =∫μ0i2πaxl2x+l2vda =μ0iv2πlnx+l2lnxl2 =μ0iv2πlnx+l2xl2
Page No 311:
Question 56:
Consider a situation similar to that of the previous problem except that the ends of the rod slide on a pair of thick metallic rails laid parallel to the wire. At one end the rails are connected by resistor of resistance R. (a) What force is needed to keep the rod sliding at a constant speed v? (b) In this situation what is the current in the resistance R? (c) Find the rate of heat developed in the resistor. (d) Find the power delivered by the external agent exerting the force on the rod.
Answer:
(a) Here, the magnetic field
B→due to the long wire varies along the length of the rod.
We will consider a small element of the rod of length da at a distance a from the wire. The magnetic field at a distance a is given by
B→=μ0i2πaNow,
Induced emf in the rod:
de=Bvdade=μ0i2πa×v× daIntegrating
xl2and
x+l2, we get
e=∫xl2x+l2de =∫μ0i2πaxl2x+l2vda =μ0iv2πlnx+l2lnxl2 =μ0iv2πlnx+l2xl2Emf induced in the rod due to the currentcarrying wire:
e=μ0iv2πln2x+l2xlNow, let the current produced in the circuit containing the rod and the resistance be i’.
i’=eR =μ0iv2πRln2x+12x1Force on the element:
dF = i’Bl
⇒dF=μ0iv2πRln2x+l2xl×μ0i2πa×da =μ0i2π2vRln2x+l2xldxaAnd,F=μ0i2π2vRln2x+l2xl∫xl/2x+l/2daa =μ0i2π2vRln2x+l2xlln2x+l2xl =vRμ0i2πln2x+l2xl2(b) Current, i‘ =
eR=μ0iv2πRln2x+l2xl(c) The rate of heat, that is, power, developed is given by
w = i^{2} R
w=μ0iv2πRln2x+l2xl2R =1Rμ0iv2πln2x+l2xl2(d) Power delivered by the external agency is the same as the rate of heat developed.
Here,
p = i^{2}R
=1Rμ0iv2πln2x+l2xl2
Question 57:
Figure shows a square frame of wire having a total resistance r placed coplanarly with a long, straight wire. The wire carries a current i given by i = i_{0} sin ωt. Find (a) the flux of the magnetic field through the square frame, (b) the emf induced in the frame and (c) the heat developed in the frame in the time interval 0 to
20πω.
Figure
Answer:
Let us consider an element of the loop of length dx at a distance x from the wire.
(a) Area of the element of loop A = adx
Magnetic field at a distance x from the wire,
B=μ0i2πxThe magnetic flux of the element is given by
dϕ=μ0i×adx2πxThe total flux through the frame is given by
ϕ=∫dϕ =∫ba+bμ0iadx2πx =μ0ia2πln1+ab(b) The emf induced in the frame is given by
e=dϕ dt =ddtμ0ia2πln1+ab =μ0a2πln1+abddt(i0 sin ωt) =μ0ai0ω cos ωt2πln1+ab(c) The current through the frame is given by
i=er =μ0ai0ω cos ωt2πrln1+abThe heat developed in the frame in the given time interval can be calculated as:
H=i2rt =μ0ai0ω cos ωt2πrln1+ab2×r×tUsing t=20πω, we getH=μ02×i2×ω24π×r2ln21+ab×r×20πω =5μ02a2i02ω2πrln21+ab
Question 58:
A rectangular metallic loop of length l and width b is placed coplanarly with a long wire carrying a current i (figure). The loop is moved perpendicular to the wire with a speed v in the plane containing the wire and the loop. Calculate the emf induced in the loop when the rear end of the loop is at a distance a from the wire. solve by using Faraday’s law for the flux through the loop and also by replacing different segments with equivalent batteries.
Figure
Answer:
Consider an element of the loop of length dx at a distance x from the currentcarrying wire.
The magnetic field at a distance x from the the currentcarrying wire is given by
B =
μ0i2πxArea of the loop = bdx
Magnetic flux through the loop element:
dϕ=μ0i2πxbdxThe magnetic flux through the loop is calculated by integrating the above expression.
Thus, we have
ϕ=∫aa+lμ0i2πxbdx =μ0i2πb∫aa+ldxx =μ0i2πxlna+laThe emf can be calculated as:
e=dϕdt=ddtμ0ib2πloga+la =μ0ib2πaa+lva(a+l) va2 ∵dadt=v =μ0ib2πaa+lvla2 =μ0ib vl2π (a+l)aCalculation of the emf using the emf method:
The emf. induced in AB and CD due to their motion in the magnetic field are opposite to each other.
The magnetic field at AB is given by
BAB=μ0i2πaNow,
Length = b
Velocity = v
The emf induced in AB is given by
eAB=μ0ivb2πaThe magnetic field at CD is given by
BCD=μ0i2π (a+l)The emf induced in side CD is given by
eCD=μ0ibv2π (a+l)The net emf induced is given by
enet=μ0ibv2πaμ0ibv2π (a+l) =μ0ibl(a+l)μibva2πa(a+l) =μ0ibvl2πa(a+l)
Question 59:
Figure shows a conducting circular loop of radius a placed in a uniform, perpendicular magnetic field B. A thick metal rod OA is pivoted at the centre O. The other end of the rod touches the loop at A. The centre O and a fixed point C on the loop are connected by a wire OC of resistance R. A force is applied at the middle point of the rod OA perpendicularly, so that the rod rotates clockwise at a uniform angular velocity ω. Find the force.
Figure
Answer:
Calculation of the emf induced in the rotating rod:
It is given that the angular velocity of the disc is ω and the magnetic field perpendicular to the disc is having magnitude B.
Let us take an element of the rod of thickness dr at a distance r from the centre.
Now,
Linear speed of the element at r from the centre, v = ωr
de=Blvde=B×dr×ωr⇒e=∫0aBωrdr =12Bωa2Because it is connected to resistance R, the current in the circuit containing the rod, wire and circular loop is given by
i=Ba2ω2RThe direction of the current is from point A to point O in the rod.
The magnitude of the force that is applied on the rod is given by
F=ilB =Ba2ω2R×a×B =B2a2ω2R
Question 60:
Consider the situation shown in the figure of the previous problem. Suppose the wire connecting O and C has zero resistance but the circular loop has a resistance R uniformly distributed along its length. The rod OA is made to rotate with a uniform angular speed ω as shown in the figure. Find the current in the rod when ∠ AOC = 90°.
Answer:
Calculation of the emf induced in the rotating rod:
It is given that the angular velocity of the disc is ω and the magnetic field perpendicular to the disc is having magnitude B.
Let us take an element of the rod of thickness dr at a distance r from the centre.
Now,
Linear speed of the element at r from the centre, v = ωr
de=Blvde=B×dr×ωr⇒e=∫0aBωrdr =12Bωa2
As ∠AOC = 90°, the minor and major segments of AC are in parallel with the rod.
The resistances of the segments are
R4and
3R4.
The equivalent resistance is given by
R’=R4×3R4R=3R16The motional emf induced in the rod rotating in the clockwise direction is given by
e=12Bωa2The current through the rod is given by
i=eR’ =Ba2ω2R’ =Ba2ω2×3R/16 =Ba2ω×162×3R=8Ba2ω3R
Question 61:
Consider a variation of the previous problem (figure). Suppose the circular loop lies in a vertical plane. The rod has a mass m. The rod and the loop have negligible resistances but the wire connecting O and C has a resistance R. The rod is made to rotate with a uniform angular velocity ω in the clockwise direction by applying a force at the midpoint of OA in a direction perpendicular to it. Find the magnitude of this force when the rod makes an angle θ with the vertical.
Answer:
When the circular loop is in the vertical plane, it tends to rotate in the clockwise direction because of its weight.
Let the force applied be F and its direction be perpendicular to the rod.
The component of mg along F is mg sin θ.
The magnetic force is in perpendicular and opposite direction to mg sin θ.
Now,
Current in the rod:
i=Ba2ω2RThe force on the rod is given by
FB=iBl=B2a2ω2RNet force = F−
B2a2ω2R+ mg sin θ
The net force passes through the centre of mass of the rod.
Net torque on the rod about the centre O:
τ=FB2a3ω2R+mg sinθOA2
Because the rod rotates with a constant angular velocity, the net torque on it is zero.
i.e.
τ=0
FB2a3ω2R+mg sinθOA2=0
∴
F=B2a3ω2Rmg sinθ
Question 62:
Figure shows a situation similar to the previous problem. All parameters are the same except that a battery of emf ε and a variable resistance R are connected between O and C. Neglect the resistance of the connecting wires. Let θ be the angle made by the rod from the horizontal position (show in the figure), measured in the clockwise direction. During the part of the motion 0 < θ < π/4 the only forces acting on the rod are gravity and the forces exerted by the magnetic field and the pivot. However, during the part of the motion, the resistance R is varied in such a way that the rod continues to rotate with a constant angular velocity ω. Find the value of R in terms of the given quantities.
Figure
Answer:
It is given that the rod is rotated with angular speed in clockwise direction.
The emf induced in the rod (e) is
Bωa22, with O at the lower potential and A at the higher potential.
The equivalent circuit can be drawn as:
i=e+ER=12Bωa2+ER=Bωa2+2E2RBecause the rod rotates with uniform angular velocity, the net torque about point O is zero.
Now,
Net force on the rod, F_{net} = mg cos θ
–ilB
Net torque, τ = (mg cos θ
–ilB).(r/2) = 0
∴ mg cos θ = ilB
R=(Bωa2+2E)2R(a×B)⇒R=(Bωa2+2E)aB2mg cos θ
Question 63:
A wire of mass m and length l can slide freely on a pair of smooth, vertical rails (figure). A magnetic field B exists in the region in the direction perpendicular to the plane of the rails. The rails are connected at the top end by a capacitor of capacitance C. Find the acceleration of the wire neglecting any electric resistance.
Answer:
Let the velocity of the rod at an instant be v and the charge on the capacitor be q.
The emf induced in the rod is given by
e = Blv
The potential difference across the terminals of the capacitor and the ends of the rod must be the same, as they are in parallel.
∴
qC=BlvAnd,
q = C × Blv = CBlv
Current in the circuit:
i =
dqdt=d(CBlv)dt
⇒i=CBldvdt=CBla (a = acceleration of the rod)
The force on the rod due to the magnetic field and its weight are in opposite directions.
∴ mg − ilB = ma
⇒ mg − cBla × lB = ma
⇒ ma + cB^{2}l^{2}a = mg
⇒ a(m + cB^{2}l^{2}) = mg
⇒a=mgm+cB2l2
Question 64:
A uniform magnetic field B exists in a cylindrical region, shown dotted in figure. The magnetic field increases at a constant rate
dBdt. Consider a circle of radius r coaxial with the cylindrical region. (a) Find the magnitude of the electric field E at a point on the circumference of the circle. (b) Consider a point P on the side of the square circumscribing the circle. Show that the component of the induced electric field at P along ba is the same as the magnitude found in part (a)
Figure
Answer:
(a) The emf induced in the circle is given by
e=dϕdt=d(B.A)dt =AdBdtThe emf induced can also be expressed in terms of the electric field as:
E.dl = e
For the circular loop,
A=πr2
⇒E2πr=πr2dBdtThus, the electric field can be written as:
E=πr22πrdBdt=r2dBdt(b) When the square is considered:
E.dl = e
For the square loop,
A=2r2
⇒E×2r×4=dBdt(2r)2⇒E=dBdt4r28r⇒E=r2dBdtThe electric field at the given point has the value same as that in the above case.
Page No 312:
Question 65:
The current in an ideal, long solenoid is varied at a uniform rate of 0.01 As^{−1}. The solenoid has 2000 turns/m and its radius is 6.0 cm. (a) Consider a circle of radius 1.0 cm inside the solenoid with its axis coinciding with the axis of the solenoid. Write the change in the magnetic flux through this circle in 2.0 seconds. (b) Find the electric field induced at a point on the circumference of the circle. (c) Find the electric field induced at a point outside the solenoid at a distance 8.0 cm from its axis.
Answer:
Given:
Rate of change of current in the solenoid,
didt= 0.01 A/s for 2s
∴didt= 0.02 A/s
n = 2000 turns/m
R = 6.0 cm = 0.06 m
r = 1 cm = 0.01 m
(a) Ï• = BA
Area of the circle, A = π × 1 × 10^{−4}
∆i=didt×∆t= (0.01 A/s) × 2 = 0.02 A/s
⇒∆ϕ=μ0nA∆iNow,
ΔÏ• = (4
π× 10^{−7}) × (2 × 10^{3}) × (
π× 10^{−4}) × (2 × 10^{−2})
= 16π^{2} × 10^{−10} Wb
= 157.91 × 10^{−10} Wb
= 1.6 × 10^{−5} Wb
∴
dϕdtfor 1s = 0.785 Wb
(b) The emf induced due to the change in the magnetic flux is given by
e=dϕdt∫Edl=dϕdt⇒E×2πr=dϕdtThe electric field induced at the point on the circumference of the circle is given by
E=0.785×1082π×102=1.2×107 V/m(c) For the point located outside the solenoid,
dϕdt=μ0ndidtA=(4π×107)×(2000)×(0.01)×(π×(0.06)2)E.dl=dϕdt⇒E=dϕ/dt2πrThe electric field induced at a point outside the solenoid at a distance of 8.0 cm from the axis is given by
E=(4π×107)×(2000)×(0.01×π×(0.06)2)(π×8×102)×dBdt=5.64×107 V/m
Question 66:
An average emf of 20 V is induced in an inductor when the current in it is changed from 2.5 A in one direction to the same value in the opposite direction in 0.1 s. Find the selfinductance of the inductor.
Answer:
Let the selfinductance of the inductor be L.
Average emf induced in the inductor, V = 20 V
Change in current, di = i_{2} − i_{1} = 2.5 − (−2.5) = 5 A
Time taken for the change, dt = 0.1 s
The voltage induced in the inductor is given by
V=Ldidt⇒20=L50.1⇒20=L×50⇒L=2050=410=0.4 H
Question 67:
A magnetic flux of 8 × 10^{−4} weber is linked with each turn of a 200turn coil when there is an electric current of 4 A in it. Calculate the selfinductance of the coil.
Answer:
Given:
Magnetic flux linked with each turn, Ï• = 8 × 10^{−4} Wb
Number of turns, n = 200
Current, i = 4 A
Selfinductance is calculated as:
L=nϕi =
2004×(8)×(104) = 4 × 10^{−2} H
Question 68:
The current in a solenoid of 240 turns, having a length of 12 cm and a radius of 2 cm, changes at a rate of 0.8 A s^{−1}. Find the emf induced in it.
Answer:
Given:
Number of turns, N = 240
Radius of the solenoid, r = 2 cm
Length of the solenoid, l = 12 cm
The emf induced in the solenoid is given by
e=LdidtThe selfinductance of the solenoid is given by
L=μ0N2AlL=4π×107×2402×π×(2×102)212×102Thus, the emf induced in the solenoid is given by
e=4π×107×2402×π×(2×102)212×102×0.8 =60577.3824×108=6×104 V
Question 69:
Find the value of t/τ for which the current in an LR circuit builds up to (a) 90%, (b) 99% and (c) 99.9% of the steadystate value.
Answer:
Current i in the LR circuit at time t is given by
i = i_{0}(1 − e^{−t}^{/τ})
Here,
i_{0} = Steadystate value of the current
(a) When the value of the current reaches 90% of the steadystate value:
i=90100×i0
90100i0= i_{o}(1 − e^{−t}^{/τ})
⇒ 0.9 = 1 − e^{−t}^{/τ}
⇒ e^{−t}^{/τ} = 0.1
On taking natural logarithm (ln) of both sides, we get
ln (e^{−t}^{/τ}) = ln 0.1
tτ=2.3⇒tτ=2.3(b) When the value of the current reaches 99% of the steadystate value:
99100i0= i_{0}(1 − e^{−t}^{/τ})
e^{−t}^{/τ} = 0.01
On taking natural logarithm (ln) of both sides, we get
ln e^{−t}^{/τ} = ln 0.01
⇒ –
tτ= − 4.6
⇒
tτ= 4.6
(c) When the value of the current reaches 99.9% of the steadystate value:
99.9100i0= i_{0}(1 − e^{−t}^{/τ})
⇒ e^{−t}^{/τ} = 0.001
On taking natural logarithm (ln) of both sides, we ge
ln e^{−t}^{/τ} = ln 0.001
⇒ ^{–}
tτ= − 6.9
⇒
tτ= 6.9
Question 70:
An inductorcoil carries a steadystate current of 2.0 A when connected across an ideal battery of emf 4.0 V. If its inductance is 1.0 H, find the time constant of the circuit.
Answer:
We know that time constant is the ratio of the selfinductance (L) of the coil to the resistance (R) of the circuit.
Given:
Current in the circuit, i = 2 A
Emf of the battery, E = 4 V
Selfinductance of the coil, L = 1 H
Now,
Resistance of the coil:
R=Ei=42=2 ΩTime constant:
τ=LR=12=0.5 s
Question 71:
A coil having inductance 2.0 H and resistance 20 â„¦ is connected to a battery of emf 4.0 V. Find (a) the current at the instant 0.20 s after the connection is made and (b) the magnetic field energy at this instant.
Answer:
Given:
Selfinductance of the coil, L = 2.0 H
Resistance in the coil, R = 20 Ω
Emf of the battery, e = 4.0 V
The steadystate current is given by
i0=eR=420A
The timeconstant is given by
τ=LR=220=0.1s
(a) Current at an instant 0.20 s after the connection is made:
i = i_{0}(1 − e^{−t}^{/τ})
=
420(1 − e^{−0.2/0.1})
=
15(1 − e^{−2})
= 0.17 A
(b) Magnetic field energy at the given instant:
12Li2=
12× 2(0.17)^{2}
= 0.0289 = 0.03 J
Question 72:
A coil of resistance 40 â„¦ is connected across a 4.0 V battery. 0.10 s after the battery is connected, the current in the coil is 63 mA. Find the inductance of the coil.
Answer:
Given:
Resistance, R = 40 â„¦
Emf of the battery, E = 4 V
Now,
The steadystate current in the LR circuit is given by
i0=440=0.1 AAt time, t = 0.1 s, the value of current i is 63 mA = 0.063 A
The current at time t is given by
i = i_{0}(1 − e^{−t}^{/τ})
⇒ 0.063 = 0.1(1 − e^{−tR}^{/L})
⇒ 63 =100(1 − e^{−4}^{/L})
⇒ 63 = 100(1 − e^{−4}^{/}^{L})
⇒ 1 − 0.63 = e^{−4}^{/L}
⇒ e^{−4}^{/L} = 0.37
⇒
4L= ln (0.37)
⇒ L =
40.994 = 4.024 H
= 4 H
Question 73:
An inductor of inductance 5.0 H, having a negligible resistance, is connected in series with a 100 â„¦ resistor and a battery of emf 2.0 V. Find the potential difference across the resistor 20 ms after the circuit is switched on.
Answer:
Given:
Selfinductance, L = 5.0 H
Resistance, R = 100 â„¦
Emf of the battery = 2.0 V
At time, t = 20 ms (after switching on the circuit)
t = 20 ms = 20 × 10^{−3} s = 2 × 10^{−2} s
The steadystate current in the circuit is given by
i0=2100The time constant is given by
τ=LR=5100s
The current at time t is given by
i = i_{0}(1 − e^{−t}^{/τ})
i=21001e2×102×1005 =2100(1e2/5) =2100(10.670) = 0.00659 = 0.0066
Now,
V = iR = 0.0066 × 100
= 0.66 V
Question 74:
The time constant of an LR circuit is 40 ms. The circuit is connected at t = 0 and the steadystate current is found to be 2.0 A. Find the current at (a) t = 10 ms (b) t = 20 ms, (c) t = 100 ms and (d) t = 1 s.
Answer:
Given:
Time constant of the given LR circuit, τ = 40 ms
Steadystate current in the circuit, i_{0} = 2 A
(a) Current at time t = 10 ms:
i = i_{0}(1 − e^{−t}^{/τ})
= 2(1 − e^{−10}^{/40})
= 2(1 − e^{−1}^{/4})
= 2(1 − 0.7788)
= 0.4422 A
= 0.44 A
(b) Current at time t = 20 ms:
i = i_{0}(1 − e^{−t}^{/τ})
= 2(1 − e^{−20}^{/40})
= 2(1 − e^{−1}^{/2})
= 2(1 − 0.606)
= 0.788 A
= 0.79 A
(c) Current at t = 100 ms:
i = i_{0}(1 − e^{−t}^{/τ})
= 2(1 − e^{−100}^{/40})
= 2(1 − e^{−10}^{/4})
= 2(1 − e^{−5}^{/2})
= 2(1−0.082)
=1.835 A
= 1.8 A
(d) Current at t = 1 s:
i = i_{0}(1 − e^{−t}^{/τ})
= 2(1 − e^{−1000}^{/40})
= 2(1 − e^{−100}^{/4})
= 2(1 − e^{−25})
= 2 × 1 A
= 2 A
Question 75:
An LR circuit has L = 1.0 H and R = 20 â„¦. It is connected across an emf of 2.0 V at t = 0. Find di/dt at (a) t = 100 ms, (b) t = 200 ms and (c) t = 1.0 s.
Answer:
Given:
Inductance, L = 1.0 H
Resistance in the circuit, R = 20 â„¦
Emf of the battery = 2.0 V
Now,
Time constant:
τ=LR=120=0.05 sSteadystate current:
i0=eR=220=0.1 ACurrent at time t:
i = i_{0}(1 − e^{−t}^{/τ})
or
i = i_{0} − i_{0}(e^{−}^{t}^{/τ})
On differentiating both sides with respect to t, we get
didt=(i0×1τet/τ) =i0τet/τ(a) At time t = 100 ms,
didt=0.10.05×e0.1/0.05=0.27 A/s(b) At time t = 200 ms,
didt=0.10.05×e0.2/0.05 =0.0366 A/s(c) At time t = 1 s,
didt=0.10.05×e1/0.05 =41×109 A/s
Question 76:
What are the values of the selfinduced emf in the circuit of the previous problem at the times indicated therein?
Answer:
Given:
Selfinductance, L = 1 H
For an inductor of selfinductance L, the emf induced is given by
e = L
didt(a) At t = 100 ms,
didt= 0.27 A/s
∴ Induced emf, e = L
didt= 1 × 0.27 = 0.27 V
(b) At t = 200 ms,
didt=0.036A/s
∴ Induced emf = L
didt= 1 × 0.036 = 0.036 V
(c) At t = 1 s,
didt= 4.1 × 10^{−9} A/s
∴ Induced emf =
Ldidt= 4.1 × 10^{−9} V
Question 77:
An inductorcoil of inductance 20 mH having resistance 10 â„¦ is joined to an ideal battery of emf 5.0 V. Find the rate of change of the induced emf at (a) t = 0, (b) t = 10 ms and (c) t = 1.0 s.
Answer:
Given:
Selfinductance, L = 20 mH
Emf of the battery, e = 5.0 V
Resistance, R = 10 â„¦
Now,
Time constant of the coil:
τ=LR=20×10310= 2 × 10^{−3} s
Steadystate current:
i0=eR=510=0.5The current in the LR circuit at time t is given by
i = i_{0}(1 − e^{−t}^{/τ})
⇒ i = i_{0} − i_{0}e^{−t}^{/τ}
On differentiating both sides, we get
didt=i0τet/τThe rate of change of the induced emf is given by
Rdidt=Ri0τ×et/τ(a) At time t = 0 s, the rate of change of the induced emf is given by
Rdidt=Ri0τ =10×0.52×103 =2.5×103 V/s(b) At time t = 10 ms, the rate of change of the induced emf is given by
Rdidt=Ri0τ×et/τNow,
For t = 10 ms = 10 × 10^{−3} s = 10^{−2} s,
dEdt=10×510×12×103×e0.01/(2×103) = 16.844 = 17 V/s
(c) At time t = 1 s, the rate of change of the induced emf is given by
dEdt=Rdidt=Ri0τ×et/τ
=10×5×1012×103×e1/(2×103) = 0.00 V/s
Question 78:
An LR circuit contains an inductor of 500 mH, a resistor of 25.0 â„¦ and an emf of 5.00 V in series. Find the potential difference across the resistor at t = (a) 20.0 ms, (b) 100 ms and (c) 1.00 s.
Answer:
Given:
Inductance of the inductor, L = 500 mH
Resistance of the resistor connected, R = 25 â„¦
Emf of the battery, E = 5 V
For the given circuit, the potential difference across the resistance is given by
V = iR
The current in the LR circuit at time t is given by
i = i_{0} (1 − e^{−tR}^{/L})
∴ Potential difference across the resistance at time t, V = (i_{0}(1 − e^{−tR}^{/L})R
(a) For t = 20 ms,
i = i_{0}(1 − e^{−tR}^{/L})
=ER(1etR/L)=525(1e(2×103×25)/(500×103)=15(1e1)=15(10.3678)=0.6325=0.1264 APotential difference:
V = iR = (0.1264) × (25)
= 3.1606 V = 3.16 V
(b) For t = 100 ms,
i = i_{0}(1 − e^{−tR}^{/L})
=5251e(100×103)×(25/500×103)=15(1e50)=15(10.0067)=0.99325=0.19864 APotential difference:
V = iR
= (0.19864) × (25) = 4.9665 = 4.97 V
(c) For t = 1 s,
i=5251e1×25/500×103 =15(1e50) =15×1=15 APotential difference:
V = iR
=15×25=5 V
Question 79:
An inductorcoil of resistance 10 â„¦ and inductance 120 mH is connected across a battery of emf 6 V and internal resistance 2 â„¦. Find the charge which flows through the inductor in (a) 10 ms, (b) 20 ms and (c) 100 ms after the connections are made.
Answer:
Given:
Inductance, L = 120 mH = 0.120 H
Resistance, R = 10 Ω
Emf of the battery, E = 6 V
Internal resistance of the battery, r = 2 Ω
The current at any instant in the LR circuit is given by
i = i_{0}(1 − e^{−t}^{/τ})
Charge dQ flown in time dt is given by
dQ = idt = i_{0}(1 − e^{−t}^{/τ})dt
Q = ∫ dQ
=∫0ti0=∫0ti0 (1et/τ)dt=i0∫0tdt∫0tet/τdt=i0t–τet/τt0=i0t+τet/τ1The steadystate current and the time constant for the given circuit are as follows:
i0=ERtotal=610+2=612=0.5 Aτ=LR=12012=0.01 s
Now,
(a) At time t = 0.01 s,
Q = 0.5 [0.01 + 0.01(e^{−0.1/0.01} − 0.01)]
= 0.00108 = 1.8 × 10^{−3} = 1.8 mΩ
(b) At t = 20 ms = 2 × 10^{−2} s = 0.02 s,
Q = 0.5 [0.02 + 0.01(e^{−0.02/0.01} − 0.01)]
= 0.005676 = 5.7 × 10^{−3} C
= 5.7 mC
(c) At t = 100 ms = 0.1 s,
Q = 0.5 [0.1 + 0.1 (e^{−0.1/0.01} − 0.01)]
= 0.045 C = 45 mc
Question 80:
An inductorcoil of inductance 17 mH is constructed from a copper wire of length 100 m and crosssectional area 1 mm^{2}. Calculate the time constant of the circuit if this inductor is joined across an ideal battery. The resistivity of copper = 1.7 × 10^{−8} â„¦m.
Answer:
Given:
Inductance, L = 17 mH
Length of the wire, l = 100 m
Crosssectional area of the wire, A = 1 mm^{2} = 1 × 10^{−6} m^{2}
Resistivity of copper, ρ = 1.7 × 10^{−8} â„¦m
Now,
R=ρlA =1.7×108×1001×106=1.7 ΩThe time constant of the LR circuit is given by
τ=LR=17×1031.7 =102 s=10 ms
Question 81:
An LR circuit having a time constant of 50 ms is connected with an ideal battery of emf ε. find the time elapsed before (a) the current reaches half its maximum value, (b) the power dissipated in heat reaches half its maximum value and (c) the magnetic field energy stored in the circuit reaches half its maximum value.
Answer:
Given:
Time constant of the LR circuit = 50 ms
Emf of the battery = ε
The time constant of the LR circuit is given by
τ=LR=50 ms=0.05 sLet the current reach half of its maximum value in time t.
Now,
i02=i0(1et/0.05)⇒12=1et/0.05⇒et/0.03=12On taking natural logarithm (ln) on both sides, we get
ln et/0.05=ln12⇒t0.05=ln(1)ln(2)⇒t0.05=00.6931⇒t=0.05×0.6931 =0.03465 s =35 ms(b) Let t be the time at which the power dissipated is half its maximum value.
Maximum power =
E2R
∴E22R=E2R(1etR/L)2⇒1eR/L=12=0.707⇒etR/L=0.293⇒t=50×1.2275 ms =61.2 ms(c) Current in the coil at the steady state, i =
εRMagnetic field energy stored at the steady state,
U=12Li2or U
=ε22R2LHalf of the value of the steadystate energy =
14Lε2R2Now,
14Lε2R2=12Lε2R2(1etR/L)2⇒etR/L=212=222⇒t=τ ln122+ln 2 =0.05ln122+ln 2 =0.061 s =61 ms
Question 82:
A coil having an inductance L and a resistance R is connected to a battery of emf ε. Find the time taken for the magnetic energy stored in the circuit to change from one fourth of the steadystate value to half of the steadystate value.
Answer:
Given:
Emf of the battery = ε
Inductance of the inductor = L
Resistance = R
Maximum current in the coil =
εRAt the steady state, current in the coil, i =
εR.
The magnetic field energy stored at the steady state is given by
U=12Li2or
U
=ε22R2LOnefourth of the steadystate value of the magnetic energy is given by
U’=18LE2R2Half of the value of the steadystate energy =
14LE2R2Let the magnetic energy reach onefourth of its steadystate value in time t_{1} and let it reach half of its value in time t_{2}.
Now,
18LE2R2=12LE2R2(1et1R/L)2⇒1et1R/L=12⇒t1RL=ln 2 And, 14LE2R2=12LE2R2(1et2R/L)2⇒et2R/L=212=222⇒t1=τ ln122+ln 2Thus, the time taken by the magnetic energy stored in the circuit to change from onefourth of its steadystate value to half of its steadystate value is given by
t2t1=τ ln122
Question 83:
A solenoid having inductance 4.0 H and resistance 10 â„¦ is connected to a 4.0 V battery at t = 0. Find (a) the time constant, (b) the time elapsed before the current reaches 0.63 of its steadystate value, (c) the power delivered by the battery at this instant and (d) the power dissipated in Joule heating at this instant.
Answer:
Given:
Inductance, L = 4.0 H
Resistance, R = 10 â„¦
Emf of the battery, E = 4 V
(a) Time constant
τ=LR=410=0.4 s(b) As the current reaches 0.63 of its steadystate value, i = 0.63 i_{0}.
Now,
0.63 i_{0} = i_{0}(1 − e^{−t}^{/τ})
⇒ e^{−t}^{/τ} = 1 − 0.063 = 0.37
⇒ ln e^{−t}^{/τ} = ln 0.37
⇒
tτ=0.9942⇒ t = 0.942 × 0.4
= 0.3977 = 0.4 s
(c) The current in the LR circuit at an instant is given by
i = i_{0}(1 − e^{−t}^{/τ})
=410(1e0.4/0.4) = 0.4 × 0.6321
= 0.2528 A
Power delivered, P = Vi
⇒ P = 4 × 0.2528
= 1.01 = 1 W
(d) Power dissipated in Joule heating, P’ = i^{2}R
⇒ P’ = (0.2258)^{2} × 10
= 0.639 = 0.64 W
Question 84:
The magnetic field at a point inside a 2.0 mH inductorcoil becomes 0.80 of its maximum value in 20 µs when the inductor is joined to a battery. Find the resistance of the circuit.
Answer:
Given:
Inductance of the inductor, L = 2.0 mH
Let the resistance in the circuit be R and the steady state value of the current be i_{0}.
At time t , current i in the LR circuit is given by
i = i_{0}(1 − e^{−t}^{/τ})
Here,
τ=LR= Time constant
On multiplying both sides by µ_{0}n, we get
n = Number of turns per unit length of the coil
µ_{0}ni = µ_{0}ni_{0}(1 − e^{−}^{t}^{/τ})
⇒ B = B_{0}(1 − e^{−tR}^{/L})
⇒ 0.8 B_{0} = B_{0}
1e20×106R2×103⇒ 0.8 = (1 − e^{−R}^{/100})
⇒ e^{−R}^{/100} = 0.2
⇒ ln (e^{−}^{R}^{/100}) = ln (0.2)
⇒
R100= −1.693
⇒ R = 169.3 â„¦
Question 85:
An LR circuit with emf ε is connected at t = 0. (a) Find the charge Q which flows through the battery during 0 to t. (b) Calculate the work done by the battery during this period. (c) Find the heat developed during this period. (d) Find the magnetic field energy stored in the circuit at time t. (e) Verify that the results in the three parts above are consistent with energy conservation.
Answer:
(a) Let the current in the LR circuit be i.
Let the charge flowing through the coil in the infinitesimal time dt be dq.
Now,
i=dqdt∴ dq = idt
The current in the LR circuit after t seconds after connecting the battery is given by
i = i_{0} (1 − e^{−t}^{/τ})
Here,
i_{0} = Steady state current
τ = Time constant =
LRdq = i_{0} (1 − e^{−tR}^{/L}) dt
On integrating both sides, we get
Q=∫0tdq
=i0∫0tdt∫0tetR/Ldt=i0t–LR(etR/L1)=i0tLR(1etR/L)Thus, the charge flowing in the coil in time t is given by
Q=εRtLR(1etR/L)
Where io = εR(b) The work done by the battery is given by
W = εQ
From the above expression for the charge in the LR circuit, we have
W=ε2RtLR(1etR/L)(c) The heat developed in time t can be calculated as follows:
H=∫0ti2 RdtH=ε2R2R∫0t(1etR/L)2dtH=ε2R∫0t(1+e2tR/L)2etR/LdtH=ε2RtL2Re2tR/L+LR2etR/L0tH=ε2RtL2Re2tR/L+LR2etR/L–L2R+2LR =ε2RtL2R(x24x+3) (x=etR/L)(d) The magnetic energy stored in the circuit is given by
U=12Li2⇒U=12Lε2R2 (1etR/L)2
⇒U=Lε22R2(1x)2(e)
Taking the sum of total energy stored in the magnetic field and the heat developed in time t
E=Lε22R2(1x)2+ε2RtL2R(x24x+3)E=Lε22R2(1+x22x)+ε2RtLε22R2(x24x+3)E=Lε22R2(2x2)+ε2RtE=Lε2R2(x1)+ε2RtE=ε2RtLR(1x)=ε2RtLR(1etR/L)The above expression is equal to the energy drawn from the battery. Therefore, the conservation of energy holds good.
Question 86:
An inductor of inductance 2.00 H is joined in series with a resistor of resistance 200 â„¦ and a battery of emf 2.00 V. At t = 10 ms, find (a) the current in the circuit, (b) the power delivered by the battery, (c) the power dissipated in heating the resistor and (d) the rate at which energy is being stored in magnetic field.
Answer:
Given:
Inductance of the inductor, L = 2 H
Resistance of the resistor connected to the inductor, R = 200 Ω,
Emf of the battery connected, E = 2 V
(a) The current in the LR circuit after t seconds after connecting the battery is given by
i = i_{0}(1 − e^{−t}^{/τ})
Here,
i_{0} = Steady state value of current
i_{0} =
ER=2200AAt time t = 10 ms, the current is given by
i
=2200(1e10×103×200/2)i = 0.01(1 − e^{−1})
i = 0.01(1 − 0.3678)
i = 0.01 × 0.632 = 6.3 mA
(b) The power delivered by the battery is given by
P = Vi
P = Ei_{0}(1 − e^{−t}^{/τ})
P=E2R(1et/τ)P=2×2200(1e10×103×200/2)P = 0.02(1 − e^{−1})
P = 0.01264 = 12.6 mW
(c) The power dissipated in the resistor is given by
P_{1} = i^{2}R
P_{1} = [i_{0}(1 − e^{−t}^{/τ})]^{2} R
P_{1} = (6.3 mA)^{2} × 200
P_{1} = 6.3 × 6.3 × 200 × 10^{−5}
P_{1} = 79.38 × 10^{−4}
P_{1} = 7.938 × 10^{−3} = 8 mW
(d) The rate at which the energy is stored in the magnetic field can be calculated as:
W =
12Li2W
=L2i02(1et/τ)2W = 2 × 10^{−2}(0.225)
W = 0.455 × 10^{−2}
W = 4.6 × 10^{−3}
W = 4.6 mW
Question 87:
Two coils A and B have inductances 1.0 H and 2.0 H respectively. The resistance of each coil is 10 â„¦. Each coil is connected to an ideal battery of emf 2.0 V at t = 0. Let i_{A} and i_{B} be the currents in the two circuit at time t. Find the ratio i_{A} / i_{B} at (a) t = 100 ms, (b) t = 200 ms and (c) t = 1 s.
Answer:
Given:
Inductance of the coil A, L_{A} = 1.0 H
Inductance of the coil B, L_{B} = 2.0 H
Resistance in each coil, R = 10 â„¦
The current in the LR circuit after t seconds after connecting the battery is given by
i = i_{0} (1 − e^{−t}^{/τ})
Here,
i_{0} = Steady state current
τ = Time constant =
LR(a) At t = 0.1 s, time constants of the coils A and B are τ_{A} and τ_{B}, respectively.
Now,
τA=110=0.1 sτB=210=0.2 sCurrents in the coils can be calculated as follows:
iA=i0 (1et/τ), =2101e0.1×101=0.2 (1e1) =0.126424111iB=i0 (1et/τ) =210(1e0.1×10/2) =0.2 (1e1/2)=0.078693 ∴iAiB=0.1264110.78693=1.6(b) At t = 200 ms = 0.2 s,
i_{A} = 0.2 (1 − e^{−0.2 × 10.1})
i_{A} = 0.2 × 0.864664716
i_{A} = 0.1729329943
i_{B} = 0.2 (1 − e^{−0.2 × 10.2})
i_{B} = 0.2 × 0.632120 = 0.126424111
∴iAiB=0.1729323430.126424111=1.36=1.4(c) At time t = 1 s,
i_{A} = 0.2 (1 − e^{−1}^{ × 10.1})
= 0.2 − 0.9999549
= 0.19999092
i_{B} = 0.2 (1 − e^{−1}^{ × 10.2})
= 0.2 × 0.99326 = 0.19865241
∴iAiB=0.199990920.19999092≈1.0
Question 88:
The current in a discharging LR circuit without the battery drops from 2.0 A to 1.0 A in 0.10 s. (a) Find the time constant of the circuit. (b) If the inductance of the circuit 4.0 H, what is its resistance?
Answer:
The current in the discharging LR circuit after t seconds is given by
i = i_{0} e^{−t}^{/τ}
Here,
i_{0} = Steady state current = 2 A
Now, let the time constant be τ.
In time t = 0.10 s, the current drops to 1 A.
i=i01et/τ⇒1=21et/τ⇒12=1et/τ⇒et/τ =112⇒et/τ = 12⇒lnet/τ = ln12=⇒0.1τ= 0.693The time constant is given by
τ=0.10.693=0.144=0.14 s
(b) Given:
Inductance in the circuit, L = 4 H
Let the resistance in the circuit be R.
The time constant is given by
τ=LRFrom the above relation, we have0.14=4R⇒R=40.14⇒R=28.5728 Ω
Page No 313:
Question 89:
A constant current exists in an inductorcoil connected to a battery. The coil is shortcircuited and the battery is removed. Show that the charge flown through the coil after the shortcircuiting is the same as that which flows in one time constant before the shortcircuiting.
Answer:
Consider an inductance L, a resitance R and a source of emf
ξare connected in series.
Time constant of this LR circuit is,
τ=LRLet a constant current i_{0} (
=ξR) is maitened in the circuit before removal of the battery.
Charge flown in one time constant before the shortcircuiting is,
Qτ=i0τ …(i)
Discahrge equation for LR circuit after short circuiting is given as,
i=i0etτChange flown from the inductor in small time dt after the short circuiting is given as,
dQ=idtChrage flown from the inductor after short circuting can be found by interating the above eqation within the proper limits of time,
Q=∫0∞idt⇒Q=∫0∞i0etτdt⇒Q=τi0etτ0∞⇒Q=τi001⇒Q=τi0 …(ii)Hence, proved.
Question 90:
Consider the circuit shown in figure. (a) Find the current through the battery a long time after the switch S is closed. (b) Suppose the switch is again opened at t = 0. What is the time constant of the discharging circuit? (c) Find the current through the inductor after one time constant.
Figure
Answer:
(a) Because the switch is closed, the battery gets connected across the Lâ€’R circuit.
The current in the Lâ€’R circuit after t seconds after connecting the battery is given by
i = i_{0} (1 − e^{−t}^{/τ})
Here,
i_{0} = Steady state current
τ = Time constant =
LRLR
After a long time, t
→∞.
Now,
Current in the inductor, i = i_{0} (1 − e^{0}) = 0
Thus, the effect of inductance vanishes.
i=εRneti=εR1×R2R1+R2=ε(R1+R2)R1R2(b) When the switch is opened, the resistance are in series.
The time constant is given by
τ=LRnet=LR1+R2(c) The inductor will discharge through resistors R_{1} and R_{2}.
The current through the inductor after one time constant is given by
t = τ
∴ Current, i = i_{0} e^{−τ}^{/τ}
Here,
i_{0} =
εR1+R2∴ i =
εR1+R2×1e
Question 91:
A current of 1.0 A is established in a tightly wound solenoid of radius 2 cm having 1000 turns/metre. Find the magnetic energy stored in each metre of the solenoid.
Answer:
Given:
Current through the solenoid, i = 1.0 A
Radius of the coil, r = 2 cm
Number of turns per metre, n = 1000
The magnetic energy density is given by
B22μ0.
Volume of the solenoid, V =
πr2lFor
l=1 m,
V=πr2.
Thus, the magnetic energy stored in volume V is given by
U =
B2πr22μ0The magnetic field is given by
B = μ_{0}ni
= (4π × 10
7) × (1000) × 1
= 4π × 10
4 T
U=(4π×104)2×4π×1042×(4π×107) =8π2×105 =78.956×105 =7.9×104 J
Question 92:
Consider a small cube of volume 1 mm^{3} at the centre of a circular loop of radius 10 cm carrying a current of 4 A. Find the magnetic energy stored inside the cube.
Answer:
Given:
Current in the loop, i = 4 A
Radius of the loop, r = 10 cm = 0.1 m
Volume of the cube, V = 1 mm^{3} =
1×109 mMagnetic field intensity at the centre of the circular loop:
B=μ0i2r =(4π×107)×42×0.1 =8π×106 TMagnetic energy density =
B22μ0Total energy stored in volume V:
U =
B2V2μ0
=(8π×106)2×(1×109)(4π×107)×2=8π×1014 J
Question 93:
A long wire carries a current of 4.00 A. Find the energy stored in the magnetic field inside a volume of 1.00 mm^{3} at a distance of 10.0 cm from the wire.
Answer:
Current flowing through the wire, i = 4.00 A
Volume of the region, V = 1 mm^{3}
Distance of the region from the wire, d = 10 cm = 0.1 m
Magnetic field due to the currentcarrying straight wire, B =
μ0i2πrThe magnetic energy stored is given by
U=B2V2μ0=μ02i24π2r2×12μ0×VU=μ0i24π2r2×12×VU=(4π×107)×(4)2×(1×109)(4π2×102)×2U=2.55×1014 J
Question 94:
The mutual inductance between two coils is 2.5 H. If the current in one coil is changed at the rate of 1 As^{−1}, what will be the emf induced in the other coil?
Answer:
Given:
Mutual inductance between the coils, M = 2.5 H
Rate of change of current in one coil,
didt= 1 As^{−1}
The flux in the coil due do another coil carrying current i is given by
Ï• = Mi
The emf induced in the second coil due to change in the current in the first coil is given by
e=dϕdt⇒e=d(Mi)dt=Mdidt⇒e=2.5×1=2.5 V
Question 95:
Find the mutual inductance between the straight wire and the square loop of figure.
Answer:
The flux through the square frame is given by
ϕ=MiLet us first calculate the flux through the square frame.
Let us now consider an element of loop of length dx at a distance x from the wire.
Now,
Area of the element of loop, A = adx
Magnetic field at a distance x from the wire,
B=μ0i2πxThe magnetic flux of the element is given by
dϕ=μ0i×adx2πxThe total flux through the frame is given by
ϕ=∫dϕ =∫ba+bμ0iadx2πx =μ0ia2πln1+abAlso,
ϕ=MiThus, the mutual inductance is calculated as
Mi=μ0ia2πln1+ab⇒M=μ0a2πln1+ab
Question 96:
Find the mutual inductance between the circular coil and the loop shown in figure.
Answer:
The magnetic field due to coil 1 at the centre of coil 2 is given by
B=μ0 Nia22 (a2+x2)3/2
The flux linked with coil 2 is given by
ϕ=B.A’=μ0 Nia22 (a2+x2)3/2πa’2Now, let y be the distance of the sliding contact from its left end.
Given:
v=dydtTotal resistance of the rheostat = R
When the distance of the sliding contact from the left end is y, the resistance of the rheostat is given by
r’=RLyThe current in the coil is the function of distance y travelled by the sliding contact of the rheostat. It is given by
i=ERLy+rThe magnitude of the emf induced can be calculated as:
e=dϕdt=μ0 Na2 a’2π 2 (a2+x2)3/2didt e=μ0 N πa2 a’22 (a2+x2)3/2ddtERLy+re=μ0 N πa2 a’22 (a2+x2)3/2ERLvRLy+r2emf induced,
e=μ0 N πa2 a’22 (a2+x2)3/2ERLvRLy+r2The emf induced in the coil can also be given as:
didt=ERLvRLy+r2
e=Mdidt , didt=ERLvRLy+r2M=edidt=Nμ0πa2a’22(a2+x2)3/2
Question 97:
A solenoid of length 20 cm, area of crosssection 4.0 cm^{2} and having 4000 turns is placed inside another solenoid of 2000 turns having a crosssectional area 8.0 cm^{2} and length 10 cm. Find the mutual inductance between the solenoids.
Answer:
Mutual inductance
M = μ_{0}N_{1}N_{2}πr_{1}^{2}l
= 4π × 10^{−7} × 4 × 10^{3} × 2 × 10^{3} × 4 × 10^{−4} × 10 × 10^{−2}
= 0.04 × 10^{−2} H
Given that,
For solenoid1
Area of cross section, a_{1} = 4 cm^{2} = 4 × 10^{−4} m^{2}
Length of the solenoid, l_{1} = 20 cm = 0.20 m
Number of turns per unit length , n_{1} = 4000/0.2 m = 20000 turns/m
For solenoid2
Area of cross section, a_{2} = 8 cm^{2} = 4 × 10^{−4} m^{2}
Length of the solenoid, l_{2} = 10 cm = 0.1 m
Number of turns per unit length, n_{2} = 2000/0.1 m = 20000 turns/m
It is given that the solenoid1 is placed inside the solenoid2
Let the current through the solenoid2 be i.
The magnetic field due to current in solenoid2
B = μ_{0}n_{2}i =
4π×107×20000×iNow,
Flux through the coil1 is given by
Ï• = n_{1}l_{1}.B.a_{1} = n_{1}l_{1}(μ_{0}n_{2}i) × a_{1}
ϕ=2000×20000×4π×107×i×4×104
If the current flowing in the coil2 changes, then emf is induced in the coil1
Thus, emf induced in the coil1 due to change in the current in coil2 is given by
Question 98:
The current in a long solenoid of radius R and having n turns per unit length is given by i = i_{0} sin ωt. A coil having N turns is wound around it near the centre. Find (a) the induced emf in the coil and (b) the mutual inductance between the solenoid ant the coil.
Answer:
Given:
Radius of the long solenoid = R
Number of turns per unit length of the long solenoid = n
Current in the long solenoid, i = i_{0} sin ωt
Number of turns in the small solenoid = N
Radius of the small solenoid = R
The magnetic field inside the long solenoid is given by
B = μ_{0}ni
Flux produced in the small solenoid because of the long solenoid, Ï• = (μ_{0}ni) × (NπR^{2})
(a) The emf developed in the small solenoid is given by
e =
dϕdt=ddt(μ0niNπR2)e = μ_{0}nN πR^{2}
didtSubstituting i = i_{0} sin ωt, we get
e = μ_{0}nNπR^{2}i_{0}ω cos ωt
(b) Let the mutual inductance of the coils be m.
Flux Ï• linked with the second coil is given by
Ï• = (μ_{0} ni) × (NπR^{2})
The flux can also be written as
Ï• = mi
∴ (μ_{0} ni) × (NπR^{2}) = mi
And,
m = πμ_{0}nNR^{2}
HC Verma Solutions for Class 11 Physics – Part 1
 Chapter 1 – Introduction to Physics
 Chapter 2 – Physics and Mathematics
 Chapter 3 – Rest and Motion: Kinematics
 Chapter 4 – The Forces
 Chapter 5 – Newton’s Laws of Motion
 Chapter 6 – Friction
 Chapter 7 – Circular Motion
 Chapter 8 – Work and Energy
 Chapter 9 – Center of Mass, Linear Momentum, Collision
 Chapter 10 – Rotational Mechanics
 Chapter 11 – Gravitation
 Chapter 12 – Simple Harmonic Motion
 Chapter 13 – Fluid Mechanics
 Chapter 14 – Some Mechanical Properties of Matter
 Chapter 15 – Wave Motion and Wave on a String
 Chapter 16 – Sound Wave
 Chapter 17 – Light Waves
 Chapter 18 – Geometrical Optics
 Chapter 19 – Optical Instruments
 Chapter 20 – Dispersion and Spectra
 Chapter 21 – Speed of Light
 Chapter 22 – Photometry
HC Verma Solutions for Class 12 Physics – Part 2
 Chapter 23 – Heat and Temperature
 Chapter 24 – Kinetic Theory of Gases
 Chapter 25 – Calorimetry
 Chapter 26 – Laws of Thermodynamics
 Chapter 27 – Specific Heat Capacities of Gases
 Chapter 28 – Heat Transfer
 Chapter 29 – Electric Field and Potential
 Chapter 30 – Gauss’s Law
 Chapter 31 – Capacitors
 Chapter 32 – Electric Current in Conductors
 Chapter 33 – Thermal and Chemical Effects of Electric Current
 Chapter 34 – Magnetic Field
 Chapter 35 – Magnetic Field due to a Current
 Chapter 36 – Permanent Magnets
 Chapter 37 – Magnetic Properties of Matter
 Chapter 38 – Electromagnetic Induction
 Chapter 39 – Alternating Current
 Chapter 40 – Electromagnetic Waves
 Chapter 41 – Electric Current through Gases
 Chapter 42 – Photoelectric Effect and Wave Particle Duality
 Chapter 43 – Bohr’s Model and Physics of the Atom
 Chapter 44 – Xrays
 Chapter 45 – Semiconductors and Semiconductor Devices
 Chapter 46 – The Nucleus
 Chapter 47 – The Special Theory of Relativity