Homework 11

Homework 11 To be submitted till 2/2/12 19:00

1 Gauge: Read the sections 10.1.2 Gauge Transformations& 10.1.3 Coulomb Gauge and Lorentz Gauge in Introduction to electrodynamics by David J. Griths. 1. Find the elds, the charge and current distributions corresponding to use the gauge function

1 qt V (r, t) = 0, A (r, t) = − 4π r. 2ˆ 0 r

1 qt − 4π to transform the potentials and comment on the result. 0 r

λ =

Now,

(Griths,

problems 10.3 & 10.5) 2. Show that it is always possible to choose

know how to solve equations of the form

∇ · A = −µ0 0 ∂V ∂t ,  2 1  V = − 0 ρ

. Is it always possible to pick

 A = 0?

as required for the Lorentz gauge, assuming you

V = 0?

How about

2 A = −µ0 J

(Griths, problem 10.7)

2 Retarded Potentials: Read the section 10.2 Continuous Distributions in Introduction to electrodynamics by David J. Griths.

J ∇· R = 1 1 0 · J) − ∇0 · 0 0 (∇ · J) + (∇ , where R = |r − r |, ∇ denotes derivatives with respect to r, and ∇ denotes R R R R 0 0 0 derivatives with respect to r . Next, noting that J r , t − c depends on r both explicitly and through R, 1˙ 1˙ 0 0 whereas it depends on r only through R, conrm that ∇ · J = − J c · (∇R), ∇ · J = −ρ˙ − c J · (∇ R). Use R 0 ´ J(r ,t− c ) µ0 this to calculate the divergence of A (r, t) = dv 0 ] (Griths, problem 10.8) 4π R

1. Conrm that the retarded potentials satisfy the Lorentz gauge condition. [Hint: First show that

J

2. An innite straight wire carries the current

I (t) = kt

for

t > 0,

where

k

is a constant.

(a) Find the electric and magnetic elds generated. (b) Do the same for the case of a sudden burst of current:

I (t) = q0 δ (t).


3 Jemenko's Equations: 1. Derive expressions for elds

ρ

r0 , t

−

R c ,

J

r0 , t

−

R c .

E

and

B

Begin with

(in integral form) as function of charge and current distributions

A (r, t) =

µ0 4π

´ J(r0 ,t− Rc ) R

dv 0

and

V (r, t) =

1 4π0

´ ρ(r0 ,t− Rc ) R

dv 0

and

show all steps of the derivation.

4 Radiation: Read the section 11.1 Dipole Radiation in Introduction to electrodynamics by David J. Griths. 1. As a model for electric quadrupole radiation, consider two oppositely oriented oscillating electric dipoles,

separated by a distance

d,

as shown in the following gure:

. Use the results of Sect.

11.1.2 (in Introduction to electrodynamics by David J. Griths) for the potentials of each dipole, but note that they are not located at the origin. Keeping only the terms of rst order in

d:

(a) Find the scalar and vector potentials. (b) Find the electric and magnetic elds. (c) Find the Poynting vector and the power radiated. Sketch the intensity prole as a function

θ.


2. A current

I (t)

ows around the circular ring in the following gure:

. Derive

´

¨2 0p the general formula for the power radiated (analogous to S · da = µ6πc ), expressing your answer in µ0 m ¨2 terms of the magnetic dipole moment (m (t)) of the loop. [Answer: P = 6πc ] . (Griths, problem 11.12)

P ∼ =

3. Point charge point charge

+q moves −q (which

in a circle (with radius

a)

with angular velocity

ω0 (r λ0 a)

is at rest in the center of the circle).

(a) Calculate the electric and magnetic dipole moments. (b) Calculate the vector potential caused by radiation. (c) Does the magnetic dipole moment contributes to the radiation? explain your answer.

around another