Classical Electrodynamics — PHY5347

HOMEWORK 22

(March 21, 2013)

Due on Thursday, April 11, 2013

PROBLEM 64

We showed in class that a time-dependent magnetic dipole generates a vector potentialat distances much larger than the dimensions of the source of the following form:

A =µ× r

cr,

where the “dot” denotes derivative with respect to the retarded time t′= t−r/c.

(a) Show that the magnetic and electric fields are given by

B =1

cA× r and E =

1

c(A× r)× r ,

and evaluate them in terms of µ and/or its time derivatives. You might finduseful to show that in this limit it is valid to replace the gradient operator by

∇ → − r

c

∂

∂t′= − r

c

∂

∂t,

whenever it acts on the vector potential.

(b) Compute, in terms of µ and/or its time derivatives, the angular distribution ofpower dP/dΩ and the total power radiated P . Recall that the angular distri-bution of power is given by

dP

dΩ= r2r · S .

(c) Assume that the permanent dipole moment of a neutron star precesses aroundthe axis of rotation of the star with frequency ω. Find the total power radiatedby the star in terms of the magnitude of µ, the magnitude of ω, and its relativeangle α. In particular, show that there will be no magnetic-dipole radiationwhen the axes are parallel to each other (i.e., when α = 0).

PROBLEM 65 (Jackson Problem 9.2)

A radiating quadrupole consists of a square of side a with charges ±q at alternatecorners. The square rotates with angular velocity ω about an axis normal to the planeof the square and through its center. Calculate in the long-wavelength approximation:

(a) The quadrupole moment.

(b) The radiation fields.

(c) The angular distribution of radiation.

(d) The total radiated power.

(e) The frequency of the emitted radiation.

Please refrain (at least initially) to look at the posted solutions that are scattered allover the Internet.

PROBLEM 66

A set of charges oscillates along the z-axis so as to produce electric quadrupole (E2)radiation. It can be shown that a quadrupole solution to the wave equation for themagnetic field may be written as B(r, t) = B(r)e−iωt, where

B(r) = f2(kr)LY20(θ, φ) (k ≡ ω/c) .

Here f2(kr) is proportional to a spherical Hankel function, L=−ir×∇ is the angularmomentum operator, and Y20(θ, φ) is a spherical harmonic.

a) Show that the magnetic field may be written as

B(r) = Cf2(kr) sin(2θ)ϕ ; C = constant

and prove that it satisfies both the wave equation and ∇ ·B=0.

b) Using the asymptotic form of the spherical Hankel function

limr→∞

f2(kr) = Aeikr

kr; A = constant ,

obtain expressions for the magnetic field B and electric field E in the radiationzone.

c) Compute the time-averaged Poynting vector and sketch the radiation (antenna)pattern. In particular, find the directions (i.e., angles) where the radiationintensity is maximized and minimized.