charge relaxation

mahinda

I have been using Born and Wolf's Principles of Optics for a project but I don't quite understand the concept of charge relaxation in metals (eq.6, P736 of 7th ed). The author used Ampere-Maxwell's law and Gauss's law to derive a differential equation involving the time differentiation of charge density and the charge density is found to fall off exponentially with time.

I don't quite understand whether this charge density refers to the charges due to polarization, or they really exponentially decay even if the metallic body carries free charges.

GRDixon

If I understand your question correctly (I don't have a copy of the reference) polarization charges are induced by an external electric field and they don't decay as long as the external field is present. Free charges within a metal decay exponentially in the process of migrating to the metal's surface. Once equilibrium has been attained, all of the excess free charge will reside on the metal object's surface. It arranges itself on the surface so that the electric field at all internal points is zero.

mahinda

Hi, thank you very much! I think I agree with you that the charge density here refers to the free charge. The author's point of mentioning the charge relaxation is that in metals Gauss's law can be considered as divergence of electric field equals zero, since any charge would exponentially decay. I'm not quite sure whether this is because the charges all migrate to the surface so that no charge exists inside the metal.

clem

The charge density referred to is free charge. A free charge distribution placed inside a conductor will decrease exponentially in magnitude, while retaining its shape until all the charge is distributed on the surface so that E will be zero inside the conductor.
There will be no charge left inside the conductor at that point.