V1.2
PHY501 Lecture III-Part 1 Multipoles and Dielectrics
Multipole Expansions
It is often useful to expand the potential due to a localized charge distribution in terms of spherical harmonics
( x) =
1
0
1 Y m( , ) qm 2 +1 r +1 ,m
y
x x'
(1)
x' R x
x
We a
Princeton University-Physics 501 Problem Set 9 Due: Dec. 20, 2011
1) A uniformly charged sphere oscillates so that the radius of the shell is an arbitrary function of time. Does it radiate? Show why or why not? 2) (J9.3) Two halves of a spherical metal sh
Solutions to Problem Set 6
1. a) The current and charge as functions of time are I(t) = I0 cos t Q(t) = Q0 sin t = I0 sin t
Electric field points in z-direction but depends just on the radial coordinate (and time), E(r, t) = E(r)^ sin t z satisfies the B
The primary text for the course will be Classical Electrodynamics (3rd Ed.), by J.D. Jackson. This book, which has long been the standard advanced treatment of E&M, is fairly formal in its approach. If you have already studies E&M at an intermediate level
V1.1
PHY501 Lecture II-Part 1 Boundary Value Problems
Boundary Value Problems
Statement of the Problem:
Find the solution to Poisson's (Laplace's) Equation
2
=
( x)
0
(1)
in a volume V bounded by a surface (or surfaces) S. The boundary conditions have two
V1.3
PHY501 Lecture II-Part 1 Boundary Value Problems
Boundary Value Problems
Statement of the Problem:
Find the solution to Poisson's (Laplace's) Equation
2
=
( x)
0
(1)
in a volume V bounded by a surface (or surfaces) S. The boundary conditions have two
V1.2
PHY501 Lecture II-Part 2 Boundary Value Problems (continued)
Spherical Harmonics
The Laplacian in spherical polar coordinates is
2
1 1 = 2 r2 + 2 r r r r sin
sin
1 + 2 2 r sin
2 2
(64)
We separate variables as before, but this time retain the possibi
Princeton University 1999
Ph501 Set 5, Solution 4
19
4. a) As the coil ips, the total amount of ux cut by the coil is 2. Then, since the
E.M.F. E generated is proportional to the rate of change of ux, the charge integrated
over time is
Q=
I dt =
1
R
1
cR
Princeton University 1999
Ph501 Set 5, Solution 1
15
The magnetic eld at the z coordinate of the particle then varies as
dBz (z(t)
dBz
=
vz .
dt
dz
(26)
If this change is slow, the analysis of part a) holds, and the particles motion varies so
2
as to keep
Princeton University 1999
Ph501 Set 5, Problem11
11
11. Consider an air-core transformer in the form of two coaxial cylinders of length l and
radii r1 < r2
l. Each cylinder is wrapped with Ni turns, and the total resistance of
coil i is Ri .
a) Deduce the
Princeton University 1999
Ph501 Set 5, Problem 9
9
9. An iron ring has a circular cross section of radius a, and average radius b
a. However,
the ring has a narrow gap from azimuth = 0 to h/b
1; the gap width is w. A
toroidal winding of N turns wraps arou
Princeton University-Physics 501 Problem Set 8 Due: Dec. 13, 2011
1) (J8.2) A transmission line consists of two concentric circular cylinders of metal with conductivity and skin depth and is filled with a uniform lossless dielectric (, ). A TEM mode is pr
Princeton University-Physics 501 Problem Set 7 Due: Dec. 6, 2011
1) (J7.4) A plane polarized electromagnetic wave of frequency in free space is incident normally on the flat surface of a nonpermeable medium of conductivity and dielectric constant . a) Cal
Princeton University-Physics 501 Problem Set 6 Due: Nov. 22, 2011
1) J6.14 (this problem is rather lengthy and will be weighted accordingly). An ideal circular parallel plate capacitor of radius a and plate separation d a is connected to a current source
V1.1
PHY501 Lecture III-Part 2 Boundary Conditions at the Interface Between Dielectrics
Dielectric Boundary Conditions
It is often useful to expand the potential due to a localized charge distribution in terms of spherical harmonics ^ Gaussian D n da = Qe
f
Leaéf F Pas/6' J:
I" km: 9|. f "h cfpl 4 F 33 {9
&./g
Q
(1
(3%..-
m g (1?fo
mmmmwmmmwmmm ..
L3 U
., f f M ,
£3 fé-Emmméi 0461M are: c79na:a'Z/CE/
6*
7161-23qu 7412. {Emaff 075 /c2 3.21:2, how Hug
a: _ I, (I H f. x 2"
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Physics 501-Fall 2011 Midterm Name: Instructions: This is an in-class closed book exam. You will find a set of potentially useful formulas at the back of the exam. There are three problems, whose weights are indicated. Partial credit will be given provide
Physics 501-Fall 2011 Midterm Name: Instructions: This is an in-class closed book exam. You will find a set of potentially useful formulas at the back of the exam. There are three problems, whose weights are indicated. Partial credit will be given provide