Chapter 2: Special Relativity

A point charge $+q$rests halfway between two steady streams of positive charge of equal charge per unit length $\lambda$, moving opposite directions and each at relative to point charge.With equal electric forces on the point charge, it would remain at rest. Consider the situation from a frame moving right at .(a) Find the charge per unit length of each stream in this frame.(b) Calculate the electric force and the magnetic force on the point charge in this frame, and explain why they must be related the way they are. (Recall that the electric field of a line of charge is $\lambda /2\pi {\epsilon }_{0}r$, that the magnetic field of a long wire is ${\mu }_{0}I/2\pi r$, and that the magnetic force is $qv\times B$. You will also need to relate $\lambda$and the current l.)

Classically, the net work done on an initially stationary object equals the final kinetic energy of the object. Verify that this also holds relativistically. Consider only one-dimension motion. It will be helpful to use the expression for p as a function of u in the following:

$W=\int Fdx=\int \frac{dp}{dt}dx=\int \frac{dx}{dt}dp=\int udp$

Both classically and relativistically, the force on an object is what causes a time rate of change of its momentum: $\mathit{F}\mathbf{=}\mathit{d}\mathit{p}\mathbf{/}\mathit{d}\mathit{t}\mathbf{.}$

(a) using the relativistically correct expression for momentum, show that

$\mathit{F}\mathbf{=}{{\mathit{\gamma }}_{\mathbf{u}}}^{\mathbf{3}}\mathit{m}\frac{du}{dt}$

(b) Under what conditions does the classical equation $\mathit{F}\mathbf{=}\mathit{m}\mathit{a}$hold?

(c) Assuming a constant force and that the speed is zero at $\mathit{t}\mathbf{=}\mathbf{0}$, separate t and u, then integrate to show that

$\mathit{u}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{+}{\left(Ft/mc\right)}^{\mathbf{2}}}}\frac{\mathbf{F}}{\mathbf{m}}\mathit{t}$

(d) Plot $\mathit{u}$verses$\mathit{t}$. What happens to the velocity of an object when a constant force is applied for an indefinite length of time?

Question: A rocket maintains a constant thrust F, giving it an acceleration of g

$\mathbf{(}\mathit{i}\mathbf{.}\mathit{e}\mathbf{.}\mathbf{,}\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}\mathbf{)}$.

(a) If classical physics were valid, how long would it take for the rocket’s speed to reach $\mathbf{0}\mathbf{.}\mathbf{99}\mathit{c}\mathbf{?}$?

(b) Using the result of exercise 117(c), how long would it really take to reach $\mathbf{0}\mathbf{.}\mathbf{99}\mathit{c}\mathbf{?}$?

$\mathit{u}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{+}{\mathbf{(}\mathbf{F}\mathbf{t}\mathbf{/}\mathbf{m}\mathbf{c}\mathbf{)}}^{\mathbf{2}}}}\frac{\mathbf{F}}{\mathbf{T}}\mathit{t}$

Exercise 117 gives the speed u of an object accelerated under a constant force. Show that the distance it travels is given by

$\mathit{x}\mathbf{=}\frac{\mathbf{m}{\mathbf{c}}^{\mathbf{2}}}{{\mathbf{f}}^{\mathbf{2}}}\left[\sqrt{1+{\left(\frac{Ft}{mc}\right)}^2}-1\right]$

Exercise 117 Gives the speed u of an object accelerated under a constant force. Show that the distance it travels is given by$x=\frac{m{c}^2}{F}\left[\sqrt{1+{\left(\frac{Ft}{mc}\right)}^2-1}\right]$.

In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length-contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about $\mathbf{30}\mathit{g}$are fatal, and there are serious concerns about the effects of prolonged accelerations greater than $\mathbf{1}\mathit{g}\mathbf{.}$ Here we see how far a person could go under a constant acceleration of 1g, producing a comfortable artificial gravity.

(a) Though traveller Anna accelerates, Bob, being on near-inertial Earth, is a reliable observer and will see less time go by on Anna's clock $\mathbf{(}\mathit{d}\mathit{t}\mathbf{\text{'}}\mathbf{)}\mathbf{}$than on his own $\mathbf{\left(}\mathit{d}\mathit{t}\mathbf{\right)}\mathbf{.}$Thus, $\mathit{d}{\mathit{t}}^{\mathbf{\text{'}}}\mathbf{\hspace{0.33em}}\mathbf{=}\mathbf{\hspace{0.33em}}\mathbf{(}\mathbf{1}\mathbf{/}\mathit{\gamma }\mathbf{)}\mathit{d}\mathit{t}$, where u is Anna's instantaneous speed relative to Bob. Using the result of Exercise $\mathbf{117}\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{,}$with $\mathit{g}$ replacing F/m, substitute for $\mathit{u},$then integrate to show that

$\mathit{t}\mathbf{=}\frac{\mathbf{c}}{\mathbf{g}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\frac{\mathbf{g}{\mathbf{t}}^{\mathbf{\text{'}}}}{\mathbf{c}}$

(b) How much time goes by for observers on Earth as they “see” Anna age 20 years?

(c) Using the result of Exercise 119, show that when Anna has aged a time t’, she is a distance from Earth (according to Earth observers) of

$\mathit{x}\mathbf{=}\frac{{\mathbf{c}}^{\mathbf{2}}}{\mathbf{g}}\left(cosh\frac{g{t}^{\text{'}}}{c}-1\right)$

(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20. How far away from Earth will she end up and how much time will have passed on Earth?

In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length-contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about $\mathbf{30}\mathbf{g}$are fatal, and there are serious concerns about the effects of prolonged accelerations greater than $\mathbf{1}\mathbf{g}$. Here we see how far a person could go under a constant acceleration of $\mathbf{1}\mathbf{g}$, producing a comfortable artificial gravity.

(a) Though traveller Anna accelerates, Bob, being on near-inertial Earth, is a reliable observer and will see less time go by on Anna's clock $\left(\mathrm{dt}\text{'}\right)$than on his own $\left(\mathrm{dt}\right)$. Thus, $\mathbf{dt}\mathbf{\text{'}}\mathbf{=}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$y$}\right.\right)\mathbf{dt}$, where $\mathbf{u}$ is Anna's instantaneous speed relative to Bob. Using the result of Exercise 117(c), with $\mathbf{g}$ replacing $\raisebox{1ex}{$\mathbf{F}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{m}$}\right.$, substitute for $\mathbf{u}$, then integrate to show that $\mathbf{t}\mathbf{=}\frac{\mathbf{c}}{\mathbf{g}}\mathbf{sinh}\frac{\mathbf{gt}\mathbf{\text{'}}}{\mathbf{c}}$.

(b) How much time goes by for observers on Earth as they “see” Anna age 20 years?

(c) Using the result of Exercise 119, show that when Anna has aged a time $\mathbf{t}\mathbf{\text{'}}$, she is a distance from Earth (according to Earth observers) of $\mathbf{x}\mathbf{=}\frac{{\mathbf{c}}^{\mathbf{2}}}{\mathbf{g}}\left(\cosh \frac{\mathrm{gt}\text{'}}{c}-1\right)$.

(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20. How far away from Earth will she end up and how much time will have passed on Earth?

Question: Two objects isolated from the rest of the universe collide and stick together. Does the system’s final kinetic energy depend on the frame of reference in which it is viewed? Does the system’s change in kinetic energy depend on the frame in which it is viewed? Explain your answer.

Appearing in the time-dilation and length-contraction formulas, ${\mathit{\gamma }}_{\mathbf{v}}$ , is a reasonable measure of the size of relativistic effects. Roughly speaking, at what speed would observations deviate from classical expectations by 1 %?