# Charged wire, relativistic fields

1. Aug 13, 2009

### eXorikos

1. The problem statement, all variables and given/known data
We have a uniformly charged wire that is infinitly long. The charge density is $$\lambda$$. Calculate the fields when the wire is at rest. Somebody moves the wire with a constant speed, what happends to the fields?

2. Relevant equations
Gauss' equation for finding the electric field at rest. Transformation rules from special relativity.

3. The attempt at a solution
The electric field is obviously:
$$E=\frac{\lambda}{2\pi r}$$
The magnetic field is zero, because there is no current.

Now we move the wire. We chose the x-axis allong the wire. y and z are perpendicular to the wire ofcourse. I assume it is ok to calculate the fields first for the case of velocity in the x-direction and then in the r-direction. The fields tranform as follows.
$$\overline{E}_x=E_x=0$$
$$\overline{B}_x=B_x=0$$
$$\overline{E}_y=\gamma (E_y-vB_z)=\gamma\frac{\lambda}{2\pi y}$$
$$\overline{E}_z=\gamma (E_z-vB_y)=\gamma\frac{\lambda}{2\pi z}$$
$$\overline{B}_y=\gamma (B_y+\frac{v}{c^2}E_z)=\gamma \frac{v}{c^2}\frac{\lambda}{2\pi z}$$
$$\overline{B}_z=\gamma (B_z-\frac{v}{c^2}E_y)=-\gamma \frac{v}{c^2}-\frac{\lambda}{2\pi y}$$

Now here's the thing. Since the observer moving allong the wire clearly sees a magnetic field caused by the passing chargedensity, according to him there must be a current in the wire. But why is the electric field in the x-direction still zero in his referenceframe? Or is it the field at the point of the observer?

How do I calculate the field for a motion in a random direction? Is it just calculating the fields for the two possible motions and adding them up or is it not that simple?