HC Verma Solutions for Class 12 Physics Chapter 47 – The Special Theory of Relativity

Page No 456:

Question 1:

The speed of light in glass is 2.0 × 108 m s−1. Does it violate the second postulate of special relativity?

Answer:

According to the second postulate of special relativity, the speed of light in vacuum has the same value c in all inertial frames.

The speed of light in glass is 2.0 × 108 m s−1, but it doesn’t violate the postulate as light follows Snell’s law, according to which the speed of light in any refractive medium gets reduced by a factor

η such that

v=cη(where

ηis the refractive index).

Thus, the speed of light in glass is 2.0 × 108 m s−1 as

ηin this case is 1.5.

Question 2:

A uniformly moving train passes by a long platform. Consider the events ‘engine crossing the beginning of the platform’ and ‘engine crossing the end of the platform’. Which frame (train frame or the platform frame) is the proper frame for the pair of events?

Answer:

The platform frame is the rest frame so it can be regarded as the proper frame for the pair of events.
The train can never approach a speed comparable to the speed of light. So, no issue of relativity comes into picture when moving train is considered as a frame for the pair of events. So, both train and platform can be taken as the proper frame for the pair of events.

Page No 457:

Question 3:

An object may be regarded to be at rest or in motion depending on the frame of reference chosen to view the object. Because of length contraction it would mean that the same rod may have two different lengths depending on the state of the observer. Is this true?

Answer:

Yes, it is true. If a rod is moving at a certain speed v, its contracted length is given as

l=lo1-v2c2, v < c where l​o is the length of the rod at rest, i.e. the length varies when measured from different frames.

Therefore, the same rod may have two different lengths depending on the state of the observer.

If the observer and the rod are moving with the same speed v in the same direction, then length of the rod l is equal to lo.
Also, if they are moving â€‹with the same speed v in the opposite direction, the measured length of the rod measured is given by

l=lo1-(v-(-v))2c2=lo1-4v2c2

Question 4:

Mass of a particle depends on its speed. Does the attraction of the earth on the particle also depend on the particle’s speed?

Answer:

Mass of a particle depends on the relativistic speed v with which it moves as

m=mo1-v2c2.
The gravitational force of attraction of Earth is given by

 

F=GMmr2⇒F=GMmor21-v2c2Note: This is the only reason responsible for gravitational lengthening in which a photon travelling at a speed c experiences gravitational force.

Question 5:

A person travelling in a fast spaceship measures the distance between the earth and the moon. Is it the same, smaller or larger than the value quoted in this book?

Answer:

The person in the spaceship can measure the distance between the Earth and the Moon either by using a metre scale or by sending a light pulse. when metre scale is used then as we know, length gets contracted in a relativistically moving frame. Therefore the distance will be smaller than the actual distance. On the other hand, on using a light  pulse and noting the time difference between its emission and reception. as we know time gets dilated in moving frame.therefore the measured difference will be larger in this case.
So, either the measured distance can be smaller or larger than the actual distance but can never be equal.

Question 1:

The magnitude of linear momentum of a particle moving at a relativistic speed v is proportional to

(a) v

(b) 1 − v2/c2

(c)

1-v2/c2(d) none of these

Answer:

(d) none of these

Linear momentum of a particle moving at a relativistic speed v is given by

p=mov1-v2c2

Here, m​o is the rest mass of the particle.

So, linear momentum is proportional to

v1-v2c2.

Hence, none of the above options is correct.

Question 2:

As the speed of a particle increases, its rest mass

(a) increases
(b) decreases
(c) remains the same
(d) changes

Answer:

(c) remains the same

Rest mass of a particle mo is universally constant. It doesn’t vary with the frames as well as with the speed with which the particle is moving.

Question 3:

An experimenter measures the length of a rod. Initially the experimenter and the rod are at rest with respect to the lab. Consider the following statements.
(A) If the rod starts moving parallel to its length but the observer stays at rest, the measured length will be reduced.
(B) If the rod stays at rest but the observer starts moving parallel to the measured length of the rod, the length will be reduced.

(a) A is true but B is false
(b) B is true but A is false
(c) Both A and B are true
(d) Both A and B are false

Answer:

(c) Both A and B are true.

If a rod is moving with speed v parallel to its length l​o and the observer is at rest, its new length will be given as

l=lo1-v2c2

As, v<c∴v2c2<1⇒1-v2c2<1Therefore, l<loIf the rod is at rest and the observer is moving with speed parallel to measured length of the rod, the rod’s length will be given as
l=lo1-(-v)2c2⇒l=lo1-v2c2As, v<c∴ -vc2<1⇒1-v2c2<1⇒l<loTherefore, the length will be reduced in both the cases.

Question 4:

An experimenter measures the length of a rod. In the cases listed, all motions are with respect to the lab and parallel to the length of the rod. In which of the cases the measured length will be minimum?

(a) The rod and the experimenter move with the same speed v in the same direction.
(b) The rod and the experimenter move with the same speed v in opposite directions.
(c) The rod moves at speed v but the experimenter stays at rest.
(d) The rod stays at rest but the experimenter moves with the speed v.

Answer:

(b) The rod and the experimenter move with the same speed v in opposite directions.

If a rod is moving with speed v parallel to its length lo and the experimenter is at rest, its new length will be given as,

l=lo1-v2c2
If the rod is at rest and the observer is moving with speed parallel to measured length of the rod, the rod’s length will be given as,
l=lo1-(-v)2c2=lo1-v2c2If the rod and the experimenter both are moving with the same speed in the same direction, then l = l​o while if they are moving with same speed in the opposite directions, the length of the rod will be given as,

l=lo1-(v-(-v))2c2=lo1-4v2c2Where, v<c​

As, 1-4v2c2<1-v2c2∴ lo1-4v2c2<lo1-v2c2<l0Therefore, the length will be minimum in the case when both are travelling in opposite direction.​

Question 5:

If the speed of a particle moving at a relativistic speed is doubled, its linear momentum will

(a) become double
(b) become more than double
(c) remain equal
(d) become less than double.

Answer:

(b) becomes more than double

If a particle is moving at a relativistic speed v, its linear momentum

pis given as,

p=mov1-v2c2⇒p=mov1-v2c2-12
Expanding binomially and neglecting higher terms we have,
p≃mov1+v22c2⇒p≃mov+mov32c2If the speed is doubled, such that it is travelling with speed 2v ,linear momentum will be given as

p’=mo(2v)1-4v2c2⇒p’=2mov1-4v2c2-12Expanding binomially and neglecting higher terms we have,

p’≃2mov1+4v22c2 ⇒p’≃2mov+4mov3c2∴  p’≃2p+3mov3c2,  3mov3c2>0Therefore, p’ is more than double of p.

Question 6:

If a constant force acts on a particle, its acceleration will

(a) remain constant
(b) gradually decrease
(c) gradually increase
(d) de undefined

Answer:

(d) gradually decrease

If a constant force is acting on a particle, the force will tend to accelerate the particle and  increase its speed. But, due to increase in speed, the mass of the particle will increase by the relation

m=mo1-v2c2 leading to decrease in acceleration as constant force is acting. Therefore, after sometime its acceleration will start decreasing gradually.

Question 7:

A charged particle is projected at a very high speed perpendicular to a uniform magnetic field. The particle will

(a) move along a circle
(b) move along a curve with increasing radius of curvature
(c) move along a curve with decreasing radius of curvature
(d) move along a straight line

Answer:

(a) move along a circle.

If a charged particle is projected at a very high speed perpendicular to a uniform magnetic field, its mass will increase from mto

m=mo1-v2c2 and its radius will increase from

Question 1:

Mark the correct statements:

(a) Equations of special relativity are not applicable for small speeds.
(b) Equations of special relativity are applicable for all speeds.
(c) Nonrelativistic equations give exact result for small speeds.
(d) Nonrelativistic equations never give exact result.

Answer:

(b) Equations of special relativity are applicable for all speeds.
(d) Nonrelativistic equations never give exact result.

According to special relativity, if a particle is moving at a very high speed v, its mass

m=γmo, length l=loγ, change in time Δt=γΔtowhere γ=11-v2c2 if v<<c⇒γ≅1that is at non-relativistic speed (small speed),

m≅mo, l≅lo, Δt≅Δto where

mo, lo and Δto are the rest mass, length and time interval respectively. Therefore, relativistic equations are applicable for all speeds. But

γ=1-v2c2-12⇒γ=1+v22c2+…   (expanding binomially)v22c2+…=k<<1 if v<<c but still k>0Hence, non relativistic equations in which

γ factor is taken to be exactly 1 never give exact results.

Question 2:

If the speed of a rod moving at a relativistic speed parallel to its length is doubled,

(a) the length will become half of the original value
(b) the mass will become double of the original value
(c) the length will decrease
(d) the mass will increase

Answer:

(c) the length will decrease
(d) the mass will increase

If the speed of a rod moving at a relativistic speed v parallel to its length, its mass 

m=γmo=mo1-v2c2 and its length

l=loγ=lo1-v2c2where γ=11-v2c2=1-v2c2-12=1+v22c2+…>1 as v<cIf the speed is doubled, its multiplying factor

γ’=11-4v2c2=1-4v2c2-12=1+2v2c2+…>2γand m=γ’mo>2γmo, l=loγ'<lo2γHence, the mass will increase but more than double and length will decrease but not exactly half of the original values.

Question 3:

Two events take place simultaneously at points A and B as seen in the lab frame. They also occur simultaneously in a frame moving with respect to the lab in a direction

(a) parallel to AB
(b) perpendicular to AB
(c) making an angle of 45° with AB
(d) making an angle of 135° with AB

Answer:

(b) perpendicular to AB

Question 4:

Which of the following quantities related to an electron has a finite upper limit?

(a) Mass
(b) Momentum
(c) Speed
(d) Kinetic energy

Answer:

(c) speed

If an electron is given a very high speed v, its mass

m=γmo=mo1-v2c2momentum, p=mv=γmov=mov1-v2c2kinetic energy, k=12mv2=12γmov2=12mov21-v2c2 γ=11-v2c2at v=c,γ=11-c2c2=∞⇒at v=c, m=p=k=∞Therefore, there’s an upper bound for v to be always less than c but no upper limits for mass, momentum and kinetic energy of the electron.

Question 5:

A rod of rest length L moves at a relativistic speed. Let L‘ = L/γ. Its length

(a) must be equal to L
(b) may be equal to L
(c) may be more than L‘ but less than L
(d) may be more than L

Answer:

(b) may be equal to L
(c) may be more than L‘ but less than L

If a rod of rest length L is moving at a relativistic speed v and its length is contracted to L‘, then

L’=Lγ=L1-v2c2If γ=11-v2c2, then v<<c, γ≅1.⇒L’≅LBut the length of the rod may be more than L‘ depending on the frame of the observer. However, it cannot be more than L because as the speed of the rod increases, its length contracts more and more due to increasing value of

γ.

Question 6:

When a rod moves at a relativistic speed v, its mass

(a) must increase by a factor of γ
(b) may remain unchanged
(c) may increase by a factor other than γ
(d) may decrease

Answer:

(a) must increase by a factor of γ

If a rod is moving at a relativistic speed v, its mass is given by

m=γmo=mo1-v2c2
Here,

γ=11-v2c2Thus, its mass must increase by a factor of

γ.

Page No 458:

Question 1:

The guru of a yogi lives in a Himalyan cave, 1000 km away from the house of the yogi. The yogi claims that whenever he thinks about his guru, the guru immediately knows about it. Calculate the minimum possible time interval between the yogi thinking about the guru and the guru knowing about it.

Answer:

Given: Distance between the house of the yogi and his guru, s = 1000 km = 106 m

So, for the minimum possible time interval, the velocity should be maximum. We know that maximum velocity can be that of light, i.e. v = 3 × 108 m/s.

We know,

Time, t=DistanceSpeed         =1063×108=1300 s

Question 2:

A suitcase kept on a shop’s rack is measured 50 cm × 25 cm × 10 cm by the shop’s owner. A traveller takes this suitcase in train moving with velocity 0.6c. If the suitcase is placed with its length along the train’s velocity, find the dimensions measured by (a) the traveller and (b) a ground observer.

Answer:

Given:
Length of suitcase, l = 50 cm
Breadth of suitcase, b = 25 cm
Height of suitcase, h = 10 cm
Velocity of train, v = 0.6c

(a) The observer in the train notices the same values of l, b and h because the suitcase is in rest w.r.t. the traveller.

(b) Since the suitcase is moving with a speed of 0.6c w.r.t. the ground observer, the component of length parallel to the velocity undergoes contraction, but the perpendicular components (breadth and height) remain the same.

So, the length that is parallel to the velocity of the train changes, while the breadth and height remain the same.

l’=l1-v2c2  =501-0.6c2c2  =501-0.36  =500.64  =50×0.8=40 cmThus, the dimensions measured by the ground observer are 40 cm × 25 cm × 10 cm.

Question 3:

The length of a rod is exactly 1 m when measured at rest. What will be its length when it moves at a speed of (a) 3 × 105 m s−1, (b) 3 × 106 m s−1 and (c) 3 × 107 m s−1?

Answer:

Given:
Proper length of the rod, L = 1 m
If v is the velocity of the rod, then the moving length of the rod is given by

L’=L1-v2c2(a) Here,
v = 3 × 105 m/s

L’=1×1-9×10109×1016    =1-10-6=0.9999995 m(b) Here,
v = 3 × 106 m/s

L’=1×1-9×10129×1016    =1-10-4    = 0.99995 m(c) Here,
v = 3 × 107 m/s

L’=1×1-9×10149×1016    =11-10-2    =0.995 m

Question 4:

A person standing on a platform finds that a train moving with velocity 0.6c takes one second to pass by him. Find (a) the length of the train as seen by the person and (b) the rest length of the train.

Answer:

Given:
Velocity of train, v = 0.6c
Time taken, t = 1 s

(a) Length observed by the observer, L= vt
⇒ L = 0.6 × 3 × 108
= 1.8 × 108 m

(b) Let the rest length of train be L0. Then,

L=L0 1-v2/c2

⇒1.8×108=L0 1-0.62 c2c2⇒1.8×108=L0×0.8⇒L0=1.8×1080.8⇒L0=2.25×108 m

Question 5:

An aeroplane travels over a rectangular field 100 m × 50 m, parallel to its length. What should be the speed of the plane so that the field becomes square in the plane frame?

Answer:

The rectangular field appears to be a square when the length becomes equal to the breadth, i.e. 50 m.
Contracted length, L‘ = 50
Original length, L = 100

Let the speed of the aeroplane be v.

Velocity of light, c = 3 × 108 m/s
We know,

L’=L1-v2/c2

⇒       50=1001-v2/c2⇒1/22=1-v2/c2⇒ 14=1-v2c2⇒ 14=c2-v2c2⇒c24=c2-v2⇒ v2=c2-c24⇒ v2=3c24⇒ v=3c2=0.866cHence, the speed of the aeroplane should be equal to 0.866c, so that the field becomes square in the plane frame.

Question 6:

The rest distance between Patna and Delhi is 1000 km. A nonstop train travels at 360 km h−1. (a) What is the distance between Patna and Delhi in the train frame?
(b) How much time elapses in the train frame between Patna and Delhi?

Answer:

Given:
Rest distance between Patna and Delhi, L0 = 1000 km = 106 m
Speed of train, v = 360 km/h

=360×518=100 m/s(a) Apparent distance between Patna and Delhi is given by

L’=L01-v2/c2L’=1061-1003×1082L’=1061-1049×1016L’=1061-19×10-12L’=56×10-9 m

Change in length = 56 nm

So, the distance between Patna and Delhi in the train frame is 56 nm less than 1000 km.

(b) Actual time taken by the train is given by

t =

L0v=1000×103100=104 s = 5003 minChange in time,

∆t=∆Lv=L’v=56×10-9100 =0.56×10-9 s=0.56 nsSo, 0.56 nm less than

5003min elapse in the train frame between Patna and Delhi.

Question 7:

A person travels by a car at a speed of 180 km h−1. It takes exactly 10 hours by his wristwatch to go from the station A to the station B. (a) What is the rest distance between the two stations? (b) How much time is taken in the road frame by the car to go from the station A to the station B?

Answer:

Given:
Speed of car, v = 180 km/hr = 50 m/s
Time, t = 10 h

Let the rest distance be L0.

Apparent distance is given by

L’=L01-v2/c2Apparent distance, L‘ = 10 × 180 = 1800 km = 18 × 105 m

Now,

18×105=L0 1-5023×1082

L0=18×105 +25×10-9 mThus, the rest distance is 25 nm more than 1800 km.

(b) Let the time taken by the car to cover the distance in the road frame be t. Then,

t=1.8×105+25×10-950 =0.36×105+5×10-8 =10 hours + 0.5 ns

Question 8:

A person travels on a spaceship moving at a speed of 5c/13. (a) Find the time interval calculated by him between the consecutive birthday celebrations of his friend on the earth. (b) Find the time interval calculated by the friend on the earth between the consecutive birthday celebrations of the traveller.

Answer:

Given: Speed of spaceship,

u=5c13(a) Proper time, t = 1 yr

Time interval calculated by him= t’=t1-u2/c2     =1 yr1-25c2169c2     =1312 yr

Thus, the time interval calculated by him between the consecutive birthday celebrations of his friend on Earth is 13/12 years.

(b) The friend on Earth also considers the same speed, so the time interval calculated by him is also 13/12 years.

Question 9:

According to the station clocks, two babies are born at the same instant, one in Howrah and other in Delhi. (a) Who is elder in the frame of 2301 Up Rajdhani Express going from Howrah to Delhi? (b) Who is elder in the frame of 2302 Dn Rajdhani Express going from Delhi to Howrah.

Answer:

The time interval recorded in a moving train, i.e. improper time is always greater than proper time (recorded in ground frame). The birth timings recorded by the station clocks is the proper time interval because it is the ground frame.
Also, the time recorded in the moving train is improper because it records time at two different places.

Hence, the proper time interval ∆T is less than that of improper, i.e. ∆T’ = v∆T.

Thus,
(a) Delhi baby is elder in the frame of 2301 Up Rajdhani Express going from Howrah to Delhi.
(b) Howrah baby is elder in the frame of 2302 Dn Rajdhani express going from Delhi to Howrah.

Question 10:

Two babies are born in a moving train, one in the compartment adjacent to the engine and other in the compartment adjacent to the guard. According to the train frame, the babies are born at the same instant of time. Who is elder according to the ground frame?

Answer:

Here, the train is in a moving frame and the clocks of the moving frame are out of synchronisation.
If L0 is the rest length separating the clocks and v is the speed of the moving frame, then the clock at the rear end leads the one at the front by L0v/c2.

Thus, the baby adjacent to the guard cell is elder to the baby adjacent to the engine.

Question 11:

Suppose Swarglok (heaven) is in constant motion at a speed of 0.9999c with respect to the earth. According to the earth’s frame, how much time passes on the earth before one day passes on Swarglok?

Answer:

Given:
Speed of Swarglok, v = 0.9999c
Proper time interval, ∆t = One day on Swarglok

Suppose ∆t‘ days pass on Earth before one day passes on Swarglok.
Now,

∆t’=∆t1-v2c2     =11-0.99992 c2c2     =10.014141782=70.712 daysThus,
∆t‘ = 70.7 days

Question 12:

If a person lives on the average 100 years in his rest frame, how long does he live in the earth frame if he spends all his life on a spaceship going at 60% of the speed of light.

Answer:

Given:
Proper time, t = 100 years
Speed of spaceship, v =

60100c= 0.60c

∆t’=∆t1-v2c2     =1001-0.602c2c2     =1000.8=125 yThus, the person lives for 125 years in the Earth frame.

Question 13:

An electric bulb, connected to a make and break power supply, switches off and on every second in its rest frame. What is the frequency of its switching off and on as seen from a spaceship travelling at a speed 0.8c?

Answer:

Given:
Speed of spaceship, v = 0.8 c
Proper frequency of bulb, f = 1 Hz

Let the frequency of bulb as seen from a spaceship be f‘.

f’=f1-v2/c2f’=1-0.64 c2c2   =0.36=0.6 Hz

Question 14:

A person travelling by a car moving at 100 km h−1 finds that his wristwatch agrees with the clock on a tower A. By what amount will his wristwatch lag or lead the clock on another tower B, 1000 km (in the earth’s frame) from the tower A when the car reaches there?

Answer:

Given:
Velocity of the car, v = 100 km/h =

100036m/s−1

Distance between tower A and tower B, s = 1000 km

If ∆t be the time interval to reach tower B from tower A, then

∆t=vs=1000100=10 h= 36000 s

∆t’=∆t1-v2/c2      =360001-100036×3×1082∆t’=360001-100036×3×1082-12∆t

Now,
∆t − ∆t‘ = 0.154 ns

∴ Time will lag by 0.154 ns.

Question 15:

At what speed the volume of an object shrinks to half its rest value?

Answer:

Let the initial volume of the object be V and the speed of the object be v respectively.

According to the question, volume of the object shrinks to half of its rest value. So apparent volume is given by

V’=V2Now, V’=V1-v2/c2⇒ V2=V1-v2/c2⇒c2=c2-v2⇒c24=c2-v2⇒v2=c2-c24=34c2⇒v=32c

Question 16:

A particular particle created in a nuclear reactor leaves a 1 cm track before decaying. Assuming that the particle moved at 0.995c, calculate the life of the particle (a) in the lab frame and (b) in the frame of the particle.

Answer:

Given:
Length of the track, d = 1 cm
Velocity of the particle, v = 0.995c

(a) Life of the particle in the lab frame is given by

t=dv=0.010.995c =0.010.995×3×108 =33.5×10-12 s=33.5 ps(b) Let the life of the particle in the frame of the particle be t‘. Thus,

t’=t1-v2/c2

t’=33.5×10-121-0.9952t’=3.3541×10-12 s=3.3541 ps

Question 17:

By what fraction does the mass of a spring change when it is compressed by 1 cm? The mass of the spring is 200 g at its natural length and the spring constant is 500 N m−1.

Answer:

Given:
Compression in the string, x = 1 cm = 1 × 10−2 m
Spring constant, k = 500 N/m
Mass of the spring, m = 200 g = 0.2 kg

Energy stored in the spring, E

=12kx2

⇒E=12×500×10-4       =0.025 JThis energy can be converted into mass according to mass energy equivalence. Thus,

Increase in mass, ∆m=Ec2=0.025c2=0.0259×1016 kgFractional change of mass, ∆mm=0.0259×1016×10.2⇒∆mm=0.01388×10-16⇒∆mm=1.4×10-18

Question 18:

Find the increase in mass when 1 kg of water is heated from 0°C to 100°C. Specific heat capacity of water = 4200 J kg−1 K−1.

Answer:

Given:
Mass of water, m = 1 kg
Specific heat capacity of water, s = 4200 J kg−1 K−1
Change in temperature, ∆θ = 100°C

Heat energy required, Q = ms∆θ
Q = 1 × 4200 × 100
= 420000 J

This energy is converted into mass. Thus,

Increase in mass of water on heating=∆m=Qc2

⇒∆m=4200003×1082          =429×1041016⇒∆m=4.66×10-12 kg≈4.7×10-12 kg

Question 19:

Find the loss in the mass of 1 mole of an ideal monatomic gas kept in a rigid container as it cools down by 100°C. The gas constant R = 8.3 J K−1 mol−1.

Answer:

Given: Number of moles of gas, n = 1

Change in temperature, ∆T = 10°C

Energy possessed by a mono atomic gas,

E=32nRdTNow,

R= 8.3 J/mol-KThis decrease in energy causes loss in mass of the gas. Thus,

Loss in mass, ∆m=Qc2 ⇒∆m=1.5×8.3×10c2          =124.59×1016=1.38×10-15 kg

Question 20:

By what fraction does the mass of a boy increase when he starts running at a speed of 12 km h−1?

Answer:

Given: Speed of the boy, v = 12 km h−1 = 10/3 m/s

Let the rest mass of the boy be m.

Kinetic energy of the boy,

E=12mv2 ⇒E=12m1032=m×509Increase in energy of the body = Kinetic energy of the boy

This increase in energy is converted into mass. Thus,

Increase in mass=∆m=Ec2∆m=m×509×9×1016Fraction increase in mass=∆mm=5081×1016⇒∆mm=5081×10-16=6.17×10-17

Question 21:

A 100 W bulb together with its power supply is suspended from a sensitive balance. Find the change in the mass recorded after the bulb remains on for 1 year.

Answer:

Given:
Power of the bulb, P = 100
W = 100 J/s

We know,
Energy = Power × Time

Hence, total energy emitted in 1 year is given by
Etotal= 100 × 3600 × 24 × 365
Etotal= 3.1536 × 109 J

This energy is converted into mass. Thus,

Increase in mass=∆m=Etotal c2=3.1536×1099×1016                       =3.504×108×10-16 kg                       =3.5×10-8 kg

Question 22:

The energy from the sun reaches just outside the earth’s atmosphere at a rate of 1400 W m−2. The distance between the sun and the earth is 1.5 × 1011 m.
(a) Calculate the rate  which the sun is losing its mass.
(b) How long will the sun last assuming a constant decay at this rate? The present mass of the sun is 2 × 1030 kg.

Answer:

Given:
Intensity of energy from Sun, I = 1400 W/m2
Distance between Sun and Earth, R = 1.5 × 1011 m

Power = Intensity × Area
P = 1400 × A
= 1400 × 4

πR2
= 1400 × 4

π× (1.5 × 1011)2
= 1400 × 4

π× (1.5)2 × 1022

Energy = Power × Time
Energy emitted in time t, E = Pt

Mass of Sun is used up to produce this amount of energy. Thus,
Loss in mass of Sun,

∆m=Ec2

⇒∆m=Ptc2⇒∆mt=Pc2             =1400×4π×2.25×10229×1016             =1400×4π×2.259×106             =4.4×109 kg/sSo, Sun is losing its mass at the rate of

4.4×109 kg/s.(b) There is a loss of 4.4 × 109 kg in 1 second. So,

2 × 1030 kg disintegrates in t‘ =

2×10304.4×109 s

⇒t’=1×10212.2×365×24×3600      =1.44×10-8×1021      =1.44×1013 y

Question 23:

An electron and a positron moving at small speeds collide and annihilate each other. Find the energy of the resulting gamma photon.

Answer:

We know,
Mass of electron = Mass of positron = 9.1 × 10−31 kg

Both are oppositely charged and annihilate each other to form a gamma photon of rest mass zero. Thus,
∆m = me + mp = 2 × 9.1 × 10−31 kg

This mass will be converted into energy of the resulting γ photon. Thus,
Eγ= ∆mc2
Eγ = 2 × 9.1 × 10−31 × 9 × 1016 J
=2×9.1×9×10-151.6×10-19=102.37×104 eV=1.02×105 eV=1.02 MeV

Question 24:

Find the mass, the kinetic energy and the momentum of an electron moving at 0.8c.

Answer:

We know,
Rest mass of electron, m0 = 9.1 × 10−31 kg
Velocity of electron, v = 0.8 c

(a) Mass of electron is given by

m=m01-v2/c2

⇒m=9.1×10-311-0.64 c2/c2=9.1×10-310.6⇒m=15.16×10-31 kg≈15.2×10-31 kg(b) Kinetic energy of electron = mc2m0c2

KE=m-m0c2     =15.2-9.1×10-31×9×1016     =5.5×10-14 J(c) Momentum of electron, p = mv

p=15.2×10-31×0.8×3×108  =3.65×1022 kgm/s

Question 25:

Through what potential difference should an electron be accelerated to give it a speed of (a) 0.6c, (b) 0.9c and (c) 0.99c?

Answer:

Kinetic energy of electron = mc2m0c2          …(1)
Suppose the electron is accelerated through a potential difference of V. Then,

KE of electron = eV

 

m=m0c21-v2/c2Putting the values of m and KE in eq. (1), we get

eV=m0c21-v2/c2-m0c2                                …(2)

(a) Velocity of electron, v = 0.6c
Rest mass of electron, m0 =

9.1×10-31 kgCharge of electron, e =

1.6×10-19 CPutting the values of m0, v and e in eq. (2), we get

eV=9.1×10-31×9×10161-0.36 c2c2-9.1×10-31×9×1016⇒V=9.1×10-31×9×10161.25-11.6×10-19⇒V=0.51×0.25 MV⇒V=0.1275 Mev= 127.5 kV(b) Putting v = 0.9c in eq. (2), we get
V = 661 kV(b) Putting v = 0.99c in eq. (2), we get
V = 3.1 MV

Question 26:

Find the speed of an electron with kinetic energy (a) 1 eV, (b) 10 keV and (c) 10 MeV.

Answer:

If m0 is the rest mass of an electron and c is the speed of light, then kinetic energy of the electron = mc2m0c2       …(1)
If

m=m0c21-v2/c2, then

(a) Kinetic energy of electron = 1 eV =

1.6×10-19 JFrom eq. (1), we get

1.6×10-19=m0c21-v2/c2-m0c2⇒1.6×10-19m0c2=11-v2/c2-1

⇒11-v/c2-1=1.6×10-199.1×10-31×9×1016⇒11-v/c2-1=0.019536×10-4⇒11-v/c2=1+0.019536×10-4⇒11-v/c2=11.0000019536⇒1-v2/c2=0.99999613⇒v2/c2=0.00000387⇒v/c=0.001967231=3×0.001967231×108         =5.92×105 m/s(b) Kinetic energy of electron = 10 keV

=1.6×10-19×10×103 J m0c211-v2/c2-1=1.6×10-15⇒9.1×10-31×9×101611-v2/c2-1=1.6×10-15⇒11-v2/c2-1=1.6×10-159.1×9×10-15⇒11-v2/c2-1=1.69.1×9⇒11-v2/c2=0.980838⇒1-v2/c2=0.962043182⇒v2/c2=1-0.962043182⇒v2=0.341611359×1018⇒v=0.584475285×108 ⇒v=5.85×107 m/s(c) Kinetic energy of electron

=10 MeV=107×1.6×10-19J

⇒m0c221-v2-c2-m0c2=1.6×10-12⇒m0c211-v2/c2-1=1.6×10-12⇒11-v2/c2-1=1.6×10-129.1×9×10-31×1016⇒11-v2/c2-1=1.019536×103⇒11-v2/c2-1=1019.536+1⇒1-v2/c2=0.000979877⇒1-v2/c2=0.99×10-6⇒v=2.999999039×108 m/s

Question 27:

What is the kinetic energy of an electron in electron volts with mass equal to double its rest mass?

Answer:

If m0 is the rest mass of an electron and c is the speed of light, then kinetic energy (E) of the electron = mc2m0c2.        …(1)

According to the question,
m = 2m0

E = (2m0m0)c2
E= m0c2 = 9.1 × 10−31 × 9 × 1016 J

E=9.1×91.6×10-1510-19=51.18×104 eV=511 keV

Question 28:

Find the speed at which the kinetic energy of a particle will differ by 1% from its nonrelativistic value

12 mov2.

Answer:

If m0 is the rest mass of a particle and c is the speed of light, then relativistic kinetic energy of particle = mc2m0c2.       …(1)

If

m=m0c21-v2/c2, then nonrelativistic kinetic energy of particle =

12 mov2.

According to the question,

m0c21-v2/c2-m0c2-12m0v21/2 m0v2=0.01⇒c21+v22c2+12×34v2c2+12×34×56v6c6-1 12v2=0.01⇒12m0v2+38m0v4/c2+1596 m0v6/c4-12m0v212m0v2=0.01⇒38 m0v2/c2+1596 m0V6/c412m0v2=0.01⇒34 v2c2+15×2v496v2c2=0.01Neglecting the v4 term as it is very small, we get

34 v2c2=0.01⇒v2c2=0.043⇒vc=0.23⇒v=223c=0.021.732×3×108=0.61.722×108 =0.345×10 m/s=3.46×107 m/s

HC Verma Solutions for Class 11 Physics – Part 1

HC Verma Solutions for Class 12 Physics – Part 2