Problems on Dynamic Fields

• Explain why we can get two apparently uncoupled wave equations for E and B from Maxwell's equations (in free space with no charges or currents). Think about the initial conditions on the wave equation.
• Explain the minus sign in Faraday's law physically.  Can you give a counterexample to exlain why it couldn't be a plus sign?
• Show that Ohm's law plus the continuity equation for charge and Maxwell's Div E equation imply a diffusion equation for chage.  For copper (or any other conductor) how long does it take before the chage decays by 1/e?  Where does the charge go (assuming the conductor is completely insulated)?
• Show that f(k.r-wt) is a traveling wave solution to the scalar, 3D wave equation where k = (kx,ky,kz) and r = (x,y,z) are vectors. What conditions on k, w, and c are required?
• Repeat the derivation of the free space wave equations for E and B but now let the charge and current be nonzero.
• Insert plane wave solutions into the wave equations for E and B and show that a) k.B and k.E = 0, and b) that k, E, and B form a orthogonal triple of vectors.
• Compute the E and B fields in the quasi-static approximation.  Show that the E field is unchanged compared to the static case but that the B field now decays like 1/r.