Chapter 12: Electrodynamics and Relativity

Inertial system $\overline{\mathbf{S}}$moves in the $x$direction at speed $\stackrel{}{\frac{\mathbf{3}}{\mathbf{5}}\mathbf{c}}$relative to system$\mathit{S}$. (The$\overline{\mathbf{x}}$axis slides long the$\mathit{x}$axis, and the origins coincide at $\mathit{t}\mathbf{=}\overline{\mathbf{t}}\mathbf{=}\mathbf{0}$, as usual.)

(a) On graph paper set up a Cartesian coordinate system with axesrole="math" localid="1658292305346" $\mathit{c}\mathit{t}$ and $\mathit{x}$. Carefully draw in lines representing$\overline{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}$and$\mathbf{3}$. Also draw in the lines corresponding to $\mathit{c}\overline{\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}$, and$\mathbf{3}$. Label your lines clearly.

(b) In$\overline{\mathbf{S}}$, a free particle is observed to travel from the point $\stackrel{}{\overline{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{,}}$at time$\mathit{c}\overline{\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{2}$to the point $\overline{\mathbf{x}}\mathbf{=}\mathbf{2}\mathbf{,}$ at$\mathit{c}\overline{\mathbf{t}}\mathbf{=}\mathbf{+}\mathbf{3}$. Indicate this displacement on your graph. From the slope of this line, determine the particle's speed in $\mathit{S}$.

(c) Use the velocity addition rule to determine the velocity in $\mathit{S}$algebraically,and check that your answer is consistent with the graphical solution in (b).

A car is traveling along the $\mathbf{45}\mathbf{°}$ line in S (Fig. 12.25), at (ordinary) speed$\mathbf{(}\mathbf{2}\mathbf{/}\sqrt{\mathbf{5}}\mathbf{)}\mathit{c}$ .

(a) Find the components${\mathbf{u}}_{\mathbf{x}}$ and${\mathbf{u}}_{\mathbf{y}}$ of the (ordinary) velocity.

(b) Find the componentsrole="math" localid="1658247416805" ${\mathit{\eta }}_{\mathbf{x}}$ and${\mathit{\eta }}_{\mathbf{y}}$ of the proper velocity.

(c) Find the zeroth component of the 4-velocity,${\mathit{\eta }}^{\mathbf{0}}$ .

System$\overline{\mathit{S}}$ is moving in the x direction with (ordinary) speed $\sqrt{\mathbf{2}\mathbf{/}\mathbf{5}}\mathit{c}$, relative to S. By using the appropriate transformation laws:

(d) Find the (ordinary) velocity components${\overline{\mathbf{u}}}_{\mathbf{x}}$ and ${\overline{\mathbf{u}}}_{\mathbf{y}}$in $\overline{\mathbf{S}}$.

(e) Find the proper velocity components ${\overline{\mathbf{\eta }}}_{\mathbf{x}}$ and${\overline{\mathbf{\eta }}}_{\mathbf{y}}$ in $\overline{\mathbf{S}}$.

(f) As a consistency check, verify that

$\stackrel{}{\overline{\mathbf{\eta }}\mathbf{=}\frac{\overline{\mathbf{u}}}{\sqrt{\mathbf{1}\mathbf{-}\frac{{\overline{\mathbf{u}}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}$

Show that it is possible to outrun a light ray, if you're given a sufficient head start, and your feet generate a constant force.

Why can’t the electric field in Fig 12.35 (b) have a, z component? After all, the magnetic field does.

A parallel-plate capacitor, at rest in ${\mathbf{S}}_{\mathbf{0}}$and tilted at a $\mathbf{45}\mathbf{°}$angle to the ${\mathit{x}}_{\mathbf{0}}$axis, carries charge densities $\mathbf{\pm }{\mathit{\sigma }}_{\mathbf{0}}$on the two plates (Fig. 12.41). System$\mathit{S}$ is moving to the right at speed $\mathit{V}$ relative to ${\mathit{S}}_{\mathbf{0}}$.

(a) Find ${\mathit{E}}_{\mathbf{0}}$, the field in ${\mathit{S}}_{\mathbf{0}}$.

(b) Find $\mathit{E}$, the field in $\mathit{S}$.

(c) What angle do the plates make with the $\mathit{x}$axis?

(d) Is the field perpendicular to the plates in $\mathit{S}$?

(a) Charge${\mathit{q}}_{\mathbf{A}}$ is at rest at the origin in system$\mathit{S}$; charge ${\mathit{q}}_{\mathbf{B}}$flies at speed$\mathit{v}$ on a trajectory parallel to the $\mathit{x}$axis, but at $\mathit{y}\mathbf{=}\mathit{d}$. What is the electromagnetic force on ${\mathit{q}}_{\mathbf{B}}$as it crosses the axis?

(b) Now study the same problem from system $\overrightarrow{\mathbf{S}}$, which moves to the right with speed . What is the force on when passes the axis? [Do it two ways: (i) by using your answer to (a) and transforming the force; (ii) by computing the fields in and using the Lorentz law.]

12.46 Two charges, $\pm q$, are on parallel trajectories a distance apart, moving with equal speeds in opposite directions. We’re interested in the force on$+q$ due to$-q$ at the instant they cross (Fig. 12.42). Fill in the following table, doing all the consistency checks you can think of as you go alon

12.48: An electromagnetic plane wave of (angular) frequency $\omega$is travelling in the $x$direction through the vacuum. It is polarized in the $y$direction, and the amplitude of the electric field is $E_o$.

(d) What is the ratio of the intensity in to the intensity in? As a youth, Einstein wondered what an electromagnetic wave would like if you could run along beside it at the speed of light. What can you tell him about the amplitude, frequency, and intensity of the wave, as approaches ?

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if ${\mathit{t}}^{\mathbf{\mu }\mathbf{v}}$is symmetric, show that${\overline{\mathbf{t}}}^{\mathbf{\mu }\mathbf{v}}$ is also symmetric, and likewise for antisymmetric).

Obtain the continuity equation (Eq. 12.126) directly from Maxwell’s equations (Eq. 12.127).