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Electromagnetic Fields and Waves: Including Electric Circuits

Paul Lorrain, Dale R. Corson

Chapter 13

Relativity I - all with Video Answers

Educators

Chapter Questions

03:42

Problem 1

(13.4) The Lorentz transformation
For what value of $\beta$ is the value of $\gamma$ equal to $1.01 ?$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:29

Problem 2

(13.4) The Lorentz transformation
Calculate $\gamma, \beta$, and $v$ for a conduction electron whose energy is 10 clectronvolts. The rest energy of an electron is $5.11 \times 10^{5}$ electronvolts.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:22

Problem 3

(13.4) Signaling problems with a fast train
Three persons $A, O^{\prime}$, and $B$, ride on a train moving at a velocity $V$, with $A$ in front, $O^{\prime}$ in the middle, and $B$ in the rear. A fourth person, $O$, stands beside the rails. At the moment $O^{\prime}$ passes $O$, light signals from $A$ and $B$ reach both $O$ and $O^{\prime}$, Persons $O$ and $O^{\prime}$ are asked who emitted her light signal first. What do they answer?

Ankur S
Ankur S
Numerade Educator
01:17

Problem 4

(13.4) Transformation of an angle
A straight line passing through the origin $O^{\prime}$ of $S^{\prime}$ forms an angle $\alpha^{\prime}$ with the $x$-axis. (a) Find a relation between $\alpha$ and $\alpha^{\prime}$.
(b) What is the value of $\alpha$ when $\mathscr{\text { tends to }} c$ ?

Dheeraj Sharma
Dheeraj Sharma
Numerade Educator
02:26

Problem 5

(13.4) Things that move faster than a photon
The Lorentz transformation implies that the relative velocity $\mathscr{V}$ of two frames of reference cannot excecd the speed of light $c$. We have also shown that a signal cannot exceed the speed of light. Discuss the following cases.
(a) $\mathrm{A}$ long, straight rod forms a small angle $\theta$ with another rod, which is horizontal and stationary. The first rod moves downward at a velocity $v$.
What is the speed of the point of intersection of the lower edge of the moving rod with the fixed rod? Can this speed be greater than $c$ ? Can the point of intersection be used to transmit a signal?
(b) The upper rod is initially at rest with the point of intersection at the origin. The rod is struck a downward blow at the origin with a hammer.
Can the motion of the point of intersection be used to transmit a signal at a speed greater than the speed of light?
(c) A powerful laser rotates rapidly about an axis perpendicular to its length.

Can the azimuthal speed of the beam cxceed the speed of light? Can the beam transmit a signal between two points at a speed greater than $c$ ?
(d) The manufacturers of some oscilloscopes claim writing speeds in excess of the speed of light. Is this possible?

Stanley Enemuo
Stanley Enemuo
Numerade Educator
02:50

Problem 6

(13.5) Space-time diagrams
Show that
$$
x^{\prime}+c t^{\prime}=\left(\frac{1-\beta}{1+\beta}\right)^{1 / 2}(x+c t)
$$
Substituting $-c$ for $c$ gives
$$
x^{\prime}-c t^{\prime}=\left(\frac{1+\beta}{1-\beta}\right)^{1 / 2}(x-c t)
$$

Vysakh M
Vysakh M
Numerade Educator
02:33

Problem 7

(13.5) $c$ is the ultimate speed
Imagine a series of reference frames $S, S^{\prime}, S^{*}, S^{w}$, etc., with $S^{\prime}$ moving at a velocity $\mathscr{V} \hat{x}$ with respect to $S, S^{\prime}$ moving at the same velocity with respect to $S^{r}$, etc. According to the Galilean transformation, a particle at rest in $S^{n}$ moves at a velocity $n \mathcal{V} \hat{x}$ with respect to $S$, where $n \mathcal{Y}$ is arbitrarily large.
Show that, according to relativity, the velocity of that particle with respect to $S$ is always less than $c$.

Adithya Ramanujam
Adithya Ramanujam
Numerade Educator
01:25

Problem 8

(13.5) Three-dimensional space-time $x, y, c t$
Fig. 13-5.
Figure $13-5$ shows the $x y$-plane, the $c t$-axis, and the planes $x=c t$ and $x=-c t$
(a) Show that a Lorentz transformation contracts this space by the factor $[(1+\beta) /(1-\beta)]^{1 / 2}$ in the direction of the plane $x=c t$, and dilates it by the same factor in the direction of $x=-c l$
(b) Show that a point $x_{11}, y_{0}, c t_{0}$ transforms to another point on the same light cone.

Nick Johnson
Nick Johnson
Numerade Educator
02:09

Problem 9

(13.5.1) The Minkowski diagram
(a) Draw a Minkowski diagram similar to that of Fig. $13-4$ with $\mathcal{V}=-0.346 \mathrm{c}$ and an event $E$ that occurs at $x=2, c t=2$.
(b) What are the values of coordinates $x^{\prime}$ and $c t^{\prime}$ ?

Juliet Schive
Juliet Schive
Numerade Educator