# Motion in a magnetic field and relativity

**1. The problem statement, all variables and given/known data**

We’re working in the right-handed Cartesian coordinate system.

Unit system is CGS.

A conducting bar of length L is placed along the x axis. Center of mass at x=0 when t=0.

It’s moving with constant velocity V in the +y direction.

There’s a uniform magnetic field B, such that:

[tex]\vec B = Bcosβ \hat y – Bsinβ \hat z[/tex]

a. Find the potential difference within the rod. Find the electric field E.

b. Find E’ and B’ in the rod’s frame of reference, far away from the rod. V=0.99c.

**2. Relevant equations**

Not given, but I assume I’ll need Lorentz force and the relativistic transformation of the fields.

**3. The attempt at a solution**

Well I think I managed a:

The moving rod’s free electrons are affected by the Lorentz force:

[tex]\vec F_{mag}=qVBsinβ(-\hat x)[/tex]

This moves the electrons to one side of the rod (the +x side) and the "positive charges" to the -x side. This causes a potential difference and an electrical field E inside the conductor.

The force due to this electric field, at least at some time t1, cancels the magnetic component of the Lorentz force:

[tex] \Sigma F_x = qE + F_{mag} = 0 [/tex]

[tex] \vec E = VBsinβ \hat x [/tex]

[tex] \Delta V = ∫\vec E.d \vec s = EL [/tex]

Now for b:

I understand that from *a* we can see that the bar is basically an electric dipole.

Being an electric dipole, it produces an electric field which I don’t know how to derive.

If I’m to derive this field, with addition to the given B field, I’ll be able to apply Lorentz transformations and get E’ and B’ outside (and far away from) the bar.

What am I missing here?

I’ve got a feeling I’m stuck due to something really dumb.

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