In Bohr’s theory of hydrogen, the electron in its ground state was supposed to travel in a circle of radius , held in orbit by the Coulomb attraction of the proton. According to classical electrodynamics, this electron should radiate, and hence spiral in to the nucleus. Show that for most of the trip (so you can use the Larmor formula), and calculate the lifespan of Bohr’s atom. (Assume each revolution is essentially circular.)

Use the result of Prob. 10.34 to determine the power radiated by an ideal electric dipole, , at the origin. Check that your answer is consistent with Eq. 11.22, in the case of sinusoidal time dependence, and with Prob. 11.26, in the case of quadratic time dependence.

Find the radiation resistance of the wire joining the two ends of the dipole. (This is the resistance that would give the same average power loss—to heat—as the oscillating dipole in fact puts out in the form of radiation.) Show that , where is the wavelength of the radiation. For the wires in an ordinary radio (say, d = 5 cm ), should you worry about the radiative contribution to the total resistance?

8 Suppose the (electrically neutral) yz plane carries a time-dependent but uniform surface current K (t) Z.

(a) Find the electric and magnetic fields at a height x above the plane if

(i) a constant current is turned on at t = 0:

(ii) a linearly increasing current is turned on at t = 0:

(b) Show that the retarded vector potential can be written in the form, and from

And from this determine E and B.

(c) Show that the total power radiated per unit area of surface is

Explain what you mean by "radiation," in this case, given that the source is not localized.22

Check that the retarded potentials of an oscillating dipole (Eqs. 11.12 and 11.17) satisfy the Lorenz gauge condition. Do not use approximation 3.

Use the “duality” transformation of Prob. 7.64, together with the fields of an oscillating electric dipole (Eqs. 11.18 and 11.19), to determine the fields that would be produced by an oscillating “Gilbert” magnetic dipole (composed of equal and opposite magnetic charges, instead of an electric current loop). Compare Eqs. 11.36 and 11.37, and comment on the result.