If a conductor or semiconductor has current flowing in it because of an applied electric field, and a transverse magnetic field is applied, there develops a component of electric field in the direction orthogonal to both the applied electric field (direction of current flow) and the magnetic field, resulting in a voltage difference between the sides of the conductor. This phenomenon is known as the Hall effects.

(a) Use the known properties of electromagnetic fields under rotations and spatial reflections and the assumption of Taylor series expansions around zero magnetic field strength to show that for an isotropic medium the generalization of Ohm's law, correct to second order in the magnetic field, must have the form

E = ρ₀J + R(H × J) + β₁H^sJ + β₂(H · J)

Where p₀ is the resistivity in the absence of the magnetic field and R is called the Hall coefficient.

(b) What about the requirements of time reversal invariance?