Electrons have magnetic dipole moment. This magnetic moment will be influenced in an electric field in case the electron get moved non-parallel to the current. The magnetic moments will be more or less aligned. During the movement of an electron in a wire under the influence of an electric potential, the electron has a chaotic movement, in addition to a drift velocity along the wire.
What is the sum of the magnetic moments after such a walk? Consider only some straight length of the wire, not the electrons "at rest" in the source and in the sink.