A2. (a) (i) Write down the expressions for the electric field E and the magnetic field B in terms of the potentials V and A in the time-independent case. (ii) Show that in the time-independent case the fields E and B satisfy the equations VXE 0 (b) (i) State how the expressions for the fields E and B in terms of potentials V and A get modified in the time-dependent case. (ii) Obtain the expressions for E and B in the time-dependent case, in terms of time-dependent potentials V and A. V.B=0 (c) (i) Explain briefly the term gauge invariance. (ii) Determine whether the time-dependent fields E and B stay invariant under the gauge transformation A new = A + Vf Vnew VAnew = - = V. - where f = f(r, t) is an arbitrary scalar field. (iii) Assuming that the potential A satisfies the Coulomb gauge condition V. A 0 t find the equation for the field f such that the gauge-transformed potentials Anew and Vnew from part (ii) satisfy the Lorentz gauge condition 1 Vnew c2 at