Jackson 7.12 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
UMASS LOWELL
C.S.BAIRD
PROBLEM: The time dependence of electrical disturbances in good conductors is governed by the frequency- dependent conductivity (7.58). Consider longitudinal electric fields in a conductor, using Ohm's law, the continuity equation, and the differential form of Coulomb's law.
(a) Show that the time-Fourier-transformed charge density satisfies the equation
[o(w)-iwe,]p(x,w)=0
(b) Using the representation (w)=./(1--i t) , where .=ew and is a damping time, show that in the approximation @, >> 1 any initial disturbance will oscillate with the plasma frequency and decay in amplitude with a decay constant 1 = 1/2. Note that if you use o() = (0) = o in part a, you will find no oscillations and extremely rapid damping with the (wrong) decay constant w = oo/eo.
SOLUTION: (a) We are assuming that the conductivity dominates so that e=eo and therefore D=e, E . The time Fourier transforms are defined according to:
p()= Jp(t)edt wherep(t)= J p(w)e-ioidw V2T V21
J()= V21
V21
E(w)= JE(t)ei"dt where E(t)=- V2T V21
at
Fourier transform the continuity equation to get it into frequency space:
V.J(w)=iwp(w)
Ohm's law states that J (w)=(w)E () . Inserting Ohm's law into the continuity equation in frequency space, we find:
o(w)VE(w)=iwp(w)
We have assumed the conductor material is spatially uniform in order to take o out of the divergence operator. Now on the right we have the divergence of the electric field, which reminds us of Coulomb's
law in differential form: V.E()=p()/e. . (Because e=eo , and there are no time operators, Coulomb's law looks the same in frequency space and time space.) Insert Coulomb's law in the equation above to find:
[o()-iw e]p()=0
(b) Now use the representation ()=./(1iw), where =ew and is a damping time.
[o(w)-iwe.]p()=0
.7
-iw|p(w)=0 1m-
If the charge density is to exist (be non-zero), then the factor in brackets must vanish.
WT 1m-
-m!+01
iv4, 21
In the approximation @,t >> 1
mf=m
Po,+if w=+wp-i/2t This tells us thatp()= Po.. if w=-w,-i/2 [0 for all other frequencies ]
Plugging this into the definition of the charge density:
p(t)=- =J p(w)e-ioidw V21
p(t)=[po,+e-io,+po.eio,]e-1/2 This tells us that an initial charge distribution will oscillate at the plasma frequency and decay with decay constant 1/