Chapter 14, Relativity Ii Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

(14.1) Lorentz contraction for a length.
A one-meter ruler moves at a speed $c / 2$. In its own reference frame it forms an angle of $45^{\circ}$ with its velocity.
What is its length, as measured by a fixed observer?

Prabhu Ramji

Numerade Educator

03:51

Problem 2

(14.1) Two successive events at a given point.
Two events occur at the same place in the laboratory at an interval of 3 seconds.

What is the spatial distance between these two events in a moving frame with respect to which the events occur 5 seconds apart, and what is the relative speed of the moving and laboratory frames?

Laszlo Zalavari

Numerade Educator

02:26

Problem 3

(14. $1.1)$ Transformation of an element of area
A small rigid parallelogram of sides $d l_{10}$ and $d l_{2 w}$, in its own reference frame, has an area
$$
d s 4_{0}=d l_{10} \times d l_{210}
$$
Show that, with respect to another reference frame, the element of area is given by
$$
d \mathscr{A}=d I_{1} \times d l_{2}=d \mathscr{}_{01}+\frac{d \mathcal{A}_{0}}{\gamma}
$$

Sriram Soundarrajan

Numerade Educator

02:33

Problem 4

(14.2) The red shift
The radio galaxy $3 \mathrm{C} 295$ has a red shift of $46 \%$. Astronomers mean by this that the observed wavelength is $1.46$ times the wavelength of the same. radiation originating in the laboratory.
(a) Calculate the radial velocity of the galaxy.
(b) Some quasars have red shifts of $200 \%$. What is their radial velocity?

Salamat Ali

Numerade Educator

05:38

Problem 5

(14.2) The case of the speeding physicist
A physicist is arrested for going through a red traffic light. In court she pleads that she approached at such a speed that the red light appeared green. The judge, a graduate of a physics class, changes the charge to speeding and fines the defendant $\$ 1$ for every kilometer per hour she exceeded the speed limit of 50 kilometers/hour.
What is the fine $\left(\lambda_{\text {gneen }}=5.3 \times 10^{-7}\right.$ meter, $\lambda_{\text {red }}=6.5 \times 10^{-7}$ meter $) ?$

Keshav Singh

Numerade Educator

01:30

Problem 6

(14.2) The twin paradox
On his twenty-first birthday, Peter leaves his twin Paul behind on the earth and goes off in a straight line for 7 years of his time at a speed of $0.96 c$. Peter then reverses direction and returns at the same speed.
(a) What are the ages of Peter and Paul at the moment of reunion?
(b) Peter and Paul, expecting a strange result, perform the following experiment during Peter's trip. They both observe a distant variable star whose light alternates from dim to bright at a frequency $f$ when observed from the earth. The variable star is in a direction perpendicular to Peter's trajectory. They, of course, both count the same number of pulsations during the trip.

Use the expression for the Doppler shift to verify the difference in age between Peter and Paul at the end of the trip. See the next problem.

Dominador Tan

Numerade Educator

01:01

Problem 7

(14.2) Doppler effect for a source moving at $y=$ constant
Figure $14-7$ shows a source of electromagnetic radiation of proper frequency $f_{0}$ situated at the origin $O^{\prime}$ of reference frame $S^{\prime}$ moving at the velocity $\mathcal{V} \hat{\mathbf{r}}$ with respect to $S$. As usual, $O^{\prime}$ is at $x=0$ at $t=0, t^{\prime}=0$. We wish to calculate the frequency as measured by observer o situated at the origin $O$ of $S$
The figure shows $S^{\prime}$ at two successive beats of the source, separated by a time
$$
t_{b}^{\prime}-t_{s}^{\prime}=\frac{1}{f_{0}}=T_{0}
$$
if $V \gamma T_{0} \ll r$
(a) Show that the period measured at $O$ is $T=\gamma(1-\beta \cos \theta) T_{0}$ and that
$$
f=f_{0} /[\gamma(1-\beta \cos \theta)]=f^{\prime} /[\gamma(1-\beta \cos \theta)]
$$
Thus, if $\theta=\pi / 2$, then $f=f_{0} / \gamma, f=f^{\prime} / \gamma$.
(b) Compare this result with the Doppler effect calculated in Sec. 14.2.1 and in Prob. 14-6.
(c) Show that
$$
f^{\prime}=\frac{f}{\gamma\left(1+\beta \cos \theta^{\prime}\right)^{-}}
$$
Thus, if $\theta^{\prime}=\pi / 2$, then $f^{\prime}=f / \gamma$ and $f=\gamma f^{\prime}$.
(d) From (a) and (c), $\gamma^{2}(1-\beta \cos \theta)\left(1+\beta \cos \theta^{\prime}\right)=1$.
Check the validity of this equation at $\theta=\theta^{\prime}=0$ and $\pi$, at $\theta=\pi / 2$, and at $\theta^{\prime}=\pi / 2$. Refer to Prob. 14-10 on the headlight effect.

Dominador Tan

Numerade Educator

01:12

Problem 8

(14.2) Transforming visible light to high-energy radiation
Visible light can be transformed into high-energy radiation by reflecting a laser beam backward on a high-energy electron beam. Say the initial photon energy is 2 electronvolts, and the electron energy is 6 gigaelectronvolts.
(a) Calculate the photon energy $h v^{\prime}$ in the reference frame $S^{\prime}$ of the electrons.
(b) Now calculate the energy $h v^{\prime \prime}$ of the reflected photons in the laboratory frame.

As a first approximation, you can neglect the recoil of the electrons, but this gives too large a value for $h v^{n}$.

Raj Bala

Numerade Educator

19:44

Problem 9

(14.4) The speed of light in a moving medium
Light moves more slowly through a material medium than through a vacuum, its phase velocity $v$ being $c / n$, where $n$ is the index of refraction of the medium.

If now the medium itself moves at a velocity $q \ll c$ with respect to the laboratory, show that the phase velocity of the light with respect to the laboratory is approximately $c / n+V\left(1-1 / n^{2}\right)$

Zachary Warner

Numerade Educator

01:59

Problem 10

(14.4) The headlight effect
A source of light moves at a velocity $V \hat{x}$. Consider a ray that forms an angle $\theta^{\prime}$ with respect to the $x$-axis.
(a) Show that in the reference frame $S, \tan \theta=\sin \theta^{\prime} /\left[\gamma\left(\cos \theta^{\prime}+\beta\right)\right]$. Then $\tan \theta^{\prime}=\sin \theta /[\gamma(\cos \theta-\beta)]$, from $\operatorname{Sec}$. 13.4
(b) Plot $\theta$ as a function of $\theta^{\prime}$ for $\beta=0,0.5,0.9,0.9999$.
Observe that, for large values of $\beta, \theta$ is much smaller than $\theta^{\prime}$, except for values of $\theta^{\text {' near } \pi \text {. If the source radiates isotropically in its own reference }} \end{array}$
frame, then, for a stationary observer, the source radiates mostly in the, forward direction. This is the headlight effect.
(c) An isotropic source of light moves at a speed $c / 3$ with respect to an observer. Calculate the solid angle defined by a cone that points forward and that contains $25 \%$ of the total light flux.

Mayukh Banik

Numerade Educator

02:26

Problem 11

$(14,4)$ The collimator paradox
Figure $14-8$ shows a source of light and a collimator $C$ fixed in a reference frame $S^{\prime}$ that moves at a velocity $V \hat{x}$ with respect to a fixed frame $S$. The detector $D$ measures the light that goes through the collimator.
15.6
According to Prob, $14-10$, the angle $\theta$ formed by the beam of light is less than $\theta^{\prime}$ because of the headlight effect. However, the Lorentz contraction on the collimator makes its angle $\theta$ larger than $\theta^{\prime}$. But this is absurd! If light reaches $D$ in $S^{\prime}$, then it does so in $S$.
You can solve this paradox by using the Lorentz transformation.
$15.8$
15.9

Stanley Enemuo

Numerade Educator

04:40

Problem 12

(14,4) Three reference frames
We have three reference frames $A, B, C$. Frames $B$ and $C$ move, respectively, at velocities $V \hat{x} / 2$ and $V \hat{x}$ with respect to $A$. Use subscripts to identify the velocities: $v_{m A}=\mathcal{V} / 2, v_{C A}=V_{\text {. }}$
Show that $v_{C A}$ (velocity of $C$ with respect to $B$ ) is larger than $V / 2$. Thus, with respect to $B, C$ moves away faster than $A$.

Subash Charan

Numerade Educator