- #1

JulienB

- 408

- 12

## Homework Statement

Hi everybody! I have the following problem to solve:

An infinitely long and thin straight wire carries a constant charge density ##\lambda## and moves at a constant relativistic speed ##\vec{v}## perpendicularly (##\beta##) and parallel (##\alpha##) to its axis.

a) Determine the ##\vec{E}## and ##\vec{B}## fields in the whole space.

b) Through which charge density ##\rho## and current density ##\vec{j}## are those fields generated?

Tip: previously we had the electrostatic potential ##\phi = - \lambda \ln(x^2+y^2)## for an infinitely long straight wire aligned with the ##z##-axis and with charge density ##\lambda##.

## Homework Equations

##\vec{E} = -\nabla \phi - \partial_t \vec{A}##

##\vec{A} = \frac{\mu_0}{4 \pi} \int d^3x'\ \frac{\vec{j} (\vec{x}', t_r)}{|\vec{x} - \vec{x}'|}##

where ##t_r = t - \frac{|\vec{x} - \vec{x}'|}{c}##

##j_{\mu} = (j^0, j^i) = (c \rho, \rho \dot{\vec{x}})##

## The Attempt at a Solution

Not sure what "parallel" and "perpendicular" here mean. Because of the given potential, I assumed that the wire is moving in the ##x##-direction at speed ##\alpha## and in the ##y##-direction at speed ##\beta##. Would you agree to this interpretation? It could also be that the wire moves in the ##z##-direction though ("parallel" to its axis), but then I imagine that the scalar potential should not be independent of ##z##. Is that right?

If my interpretation is correct I can calculate ##-\nabla \phi## which yields

##-\nabla \phi = \frac{2 \lambda}{ x^2+y^2} (x,y,0) = \frac{2 \lambda t}{t^2 (\alpha^2 + \beta^2)} (\alpha, \beta, 0)##

where I set ##x=\alpha t## and ##y=\beta t##. I am now not sure how to proceed regarding ##\vec{A}## since we didn't study the retarded time/potential yet (I could give it a try though). Should I try to calculate ##t_r## or is there another method I can use?Thanks a lot in advance for your suggestions.Julien.