Simple Nature - Light and Matter

time I used it implicitly was in figure c, where I assumed that the

velocities of the two line charges wereu−vand−u−v. Relativistic

velocities don’t just combine by simple addition and subtraction like

this, but this is an effect we can ignore in the present approximation.

The second sleight of hand occurred when I stated that we could

equate the forces in the two frames of reference. Force, like time

and distance, is distorted relativistically when we change from one

frame of reference to another. Again, however, this is an effect that

we can ignore to the desired level of approximation.

Let±λbe the charge per unit length of each line charge with-

out relativistic contraction, i.e., in the frame moving with that line

charge. Using the approximationγ= (1−v^2 /c^2 )−^1 /^2 ≈1 +v^2 / 2 c^2

forvc, the total charge per unit length in frame 2 is

λtotal, 2≈λ

[

1 +

(u−v)^2

2 c^2

]

−λ

[

1 +

(−u−v)^2

2 c^2

]

=

− 2 λuv

c^2

.

LetRbe the distance from the line charge to the lone charge. Ap-

plying Gauss’ law to a cylinder of radius Rcentered on the line

charge, we find that the magnitude of the electric field experienced

by the lone charge in frame 2 is

E=

4 kλuv

c^2 R

,

and the force acting on the lone chargeqis

F=

4 kλquv

c^2 R

.

In frame 1, the current isI = 2λ 1 u (see homework problem 5),

which we can approximate asI = 2λu, since the current, unlike

λtotal, 2, doesn’t vanish completely without the relativistic effect.

The magnetic force on the lone chargeqdue to the currentIis

F=

2 kIqv

c^2 R

.

Discussion Questions

A In the situation shown in figure c, is there a frame in which the force

Fis a purely electric one,FE? PureFB?

Is there a frame in which the electromagnetic field is a pureE? PureB?

Is the charge densityρzero in both frames? One? Neither?

What about the currentI(or current densityj)?

B For the situation shown in figure c, draw a spacetime diagram in the

style demonstrated in sec. 7.2, p. 400, showing the positive charges as

black world-lines and the negative as red, in the wire’s rest frame. Use a

ruler, and draw the spacing fairly accurately. Interpret this in the frame of

the lone charge.

676 Chapter 11 Electromagnetism