2. (a) (i) Write down Maxwell's equations for static electric and magnetic ﬁelds in the vacuum (note that you should include charge and current densities).
(ii) How did Maxwell modify Ampére's law to account for dynamic electric ﬁelds?
(b) In a region of space in which the relative permittivity is (attached equation), the relative permeability is (attached), and the free current density is J = 0, the magnetic ﬁeld, B, is given by,
Here A, k, and w are constants, t is time, x, y, and z are cartesian coordinates and i and j are unit vectors in the x and y directions respectively.
(i) Find the auxiliary ﬁeld, H.
(ii) Find the magnetization, M.
(iii) Find delta * B and explain the physical meaning of your result.
(iv) Identify the circumstances in which delta * H = - delta * M.
(v) Find delta X B.
(vi) Find delta x E.
(vii) Does E = - delta V in this case?
(viii) Find delta E / delta t.
(ix) Deduce a possible expression for the electric ﬁeld, E.
(x) Find an expression for the polarization vector, P.
(xi) Find the volume density of bound charge.
(xii) Find the volume density of free charge.
The following posting helps with several advanced problems in electromagnetism.