Applying Lorentz transform to current 4 vector

[Finch91]

Homework Statement

Consider an infinite line of charge with density λ per unit length lying along the z-axis. If the line of charge is stationary in frame S, use the Lorentz transform to find the current and charge densities in a frame S' which is moving with velocity v parallel to the z-axis of S.

Homework Equations

4 current defined as

$$j_\mu = (J_1,J_2,J_3,c\rho)$$

The Attempt at a Solution

In S I find

$$j_\mu = (0,0,0,c\lambda)$$

then applying the Lorentz transform to find

$$j'_\mu = (-\gamma\beta c \lambda,0,0,\gamma c \lambda)$$

in S'

Is this correct?

 

[BruceW]

Hey, welcome to physicsforums!
It's almost correct. You have the correct answer if the charge was uniform over all space. (since you have put the constant lambda, the charge will spread the same over all space). What the question asked for was a charge distribution that is concentrated on the z axis. So which mathematical object can you use to represent this charge distribution?

Also, usually the charge is written as the zeroth component, and the current as the other 3, but I guess it is only convention, so it doesn't really matter. It just might be confusing when you look at other people's work.