If a charge moves in an electric and in a magnetic field, it experiences a force according to the cross product,

**F**=q (**E**+ **v**x**B**),

where **F** is the force vector, q is the charge, **v** is the velocity of the charged particle, **E** is the electric field and **B** is the magnetic field. There's one other tricky bit here. Current is actually pointing in the opposite direction of electron flow because it was originally defined before people knew what was flowing in wires.

The problem is usually stated as: Imagine you have a current carrying wire which creates a magnetic field that loops around it (in the same direction as your fingers curl if you point the thumb of your right hand along the current). In this case, the wire is neutral, so it has no electric field. The force equation simplifies to only the magnetic field term:

**F**=q **v**x**B**

An electron moving next to the wire in the direction of the current experiences a force in the direction given by the cross product of its velocity vector and the magnetic field vector of the wire at its location. (If you don't know how to do a cross product, look up the right-hand-rule, since its easier to see with pictures than me explaining it in text.) The net result is a repulsive force.

If you decide to hitch a ride on the electron, the electron's velocity goes to zero, so the magnetic term in the force equation is zero. Now, the electron is moving in the same direction as the current, so it's moving opposite the electron flow. This means the negative bits of the wire are moving more quickly with respect to you than the positive bits, so you see more of a contraction of the negative bits than the positive bits, so the wire appears to be negatively charged, which sets up an electric field pointing towards the wire, which in turn repels the electron (it's negative so it tends to move in the opposite direction the electric field points.)

I know the electron drift velocity is slow, but I think the quantity of interest here is actually current flow, so relativity applies. If you want to talk about electrons, you'd probably need to deal with a more complicated quantum theory, but the same results should eventually pop out.