Problem Set 6 The charge density for a hydrogenic p state can be written as r 2 exp r [P 0 (cos ) P 2 (cos )] (r, ) = 2 3 e a0 a0 3 a 0 64 e = 4. 8 10 10 statCoulomb, a 0 = 0. 529 10 8 cm. (a) Determine the multipole moments of this charge distribution an
411 Problem Set 5 2009 1. Using the solution to Laplaces equation in cylindrical coordinates, 2 (, , z) = 0,
where (, , z) = ,m [a m J m () + b m N m ()]e im e z write a general form for the Greens function, 4 1 r , in cylindrical coordinates for all of r
411 Problem Set 4 2009 1.
Find a solution for the elecrostatic potential inside the following volume, V: -a/2 x a/2, -b/2 y b/2; z 0. Assume that there are no charges in the volume and (a/2, y, z) = (x, b/2, z) = 0
(x, y, z ) = 0 The solution will not be
411 Problem Set 3 Spring 09
of the charge stored and V is the magnitude of the electrostatic potential difference between the two surfaces which form the capacitor. (a) In each of the following find (r) or E(r) between the two surfaces and determine the c
Phys 411 Set 2
Jackson problem 1.3
Using the Dirac delta function in the appropriate coordinates express the following charge distributions as three dimensional charge densities (r ). 1. In spherical coordinates, a charge Q uniformly distributed over a sp
Phys 411 Set 1 Special problems
1. (a) By using Gauss law (Eq. ref: Gauss ) and the definition of the Dirac delta function show that if q Er = k 1 3 r r then the charge density is _r = q N 3 r . (b) Show that if _ r = and Er = k 1 X then Er = k 1 >
i=1 N
Physics 411 Exam I
March 2, 2004
Name_ 1._(50 points) 2._(25 points) 3._(25 points) total ._(/100 points)
Dr, t 4k 1 o 0 r, t Br, t 0 Er, t k 3
Dr Er
D 1 r, t D 2 r, t n r, t B 1 r, t B 2 r, t n 0 n E 1 r, t E 2 r, t 0 n H 1 r, t H 2 r, t r, t r 1 4 o
B
*Physics
411 EXAM COVER SHEET.*
You can write any formulas you like on the printed side of these sheets and bring them to the exam. _ Dr = PEr 4 6 Dr, t = 4^k 1 O o _ 0 r, t D 1 r, t ? D 2 r, t 6 n = ar, t 4 6 Br, t = 0 /Br, t /t J /Dr, t + 4^k 2 J J 0 r,
Chapter 4
Boundary Value Problems in Spherical and Cylindrical Coordinates
4.1
Laplaces Equation in spherical coordinates
Laplaces equation in spherical coordinates, (r, , ) , has the form 1 1 r2 (r, , ) 2 L L (r, , ) = 0. 2 r r r r L L 1 1 2 sin (r, , )
Section 4.9
Laplaces equation in cylindrical coordinates
As in the case of spherical coordinates, this equation is solved by a series expansion in terms of products of functions of the individual cylindrical coordinates. That is, we use separation of vari
Page 61 Solutions to Laplaces equation: (x,y,z), (r,S, j, _, j, z; Helmholtz equation in (r,S, j
Equation
4 2 x, y, z = 0
separation. const.
! ! k=k 1 x +k 2 +k 3 z
General solution: sum over all sep. constants
# #
conditions
# # F k6k =0 cke k 6 r + dke
Chapter 3
Boundary Value Problems, Introduction
3.1 The method of images There are some problems in electrostatics for which a solution can be obtained by adding image charges outside the region of interest. The most obvious one is the potential of a char
Chapter 2 Introduction to electrostatics
2.1 Coulomb and Gauss Laws
We will restrict our discussion to the case of static electric and magnetic elds in a homogeneous, isotropic medium. In this case the electric eld satises the two equations, Eq. 1.59a wit
Fields and potential due to a surface electric dipole layer
A surface electric dipole layer is a neutral charge layer with an electric dipole moment per unit area directed perpendicular to the surface. It can be modeled as two surface charge layers, (r, )
Chapter 1
Introduction and Survey
1.1 Maxwells equations in a vacuum
1.1.1 Electrostatics The results of the numerous investigations of electromagnetic phenomena carried out during the 18th and 19th centuries led to the development of a set of equations w
IV-30
The Dirac Delta Function, (x-xo)
Dirac Delta Function In one dimension, (x-xo) is defined to be such that: ma to b f(x) (x-xo)dx /
+ *0 if xo is not in [a,b]. *f(xo) if xo = a or b; *f(xo) if xo (a,b). .
Properties of (x-xo): (you should know those
Physics 411 Final
May 2002
SAMPLE
Name_ 1._(80 points) 2._(35 points) 3._(35 points) total ._(/150 points)
_ Dr = PEr 4 6 Dr, t = 4^k 1 O o _ 0 r, t D 1 r, t ? D 2 r, t 6 n = ar, t 4 6 Br, t = 0 /Br, t /t k 2 J /Dr, t + 4^k J J r, t 4 H r , t = 2W 0 o k1
Physics 411 Final
Name_
cover p_1
1._(80 points) course ave _ 2._(35 points) course grade _ 3._(35 points) total ._(/150 points) _ 4 6 Dr, t = 4^k 1 O o _ 0 r, t Dr = PEr D 1 r, t ? D 2 r, t 6 n = ar, t 4 6 Br, t = 0 /Br, t /t J /Dr, t + 4^k J J r, t 4 H
Physics 411 Final
May 2001
SAMPLE
Name_ 1._(80 points) 2._(35 points) 3._(35 points) total ._(/150 points)
_ Dr = PEr 4 6 Dr, t = 4^k 1 O o _ 0 r, t D 1 r, t ? D 2 r, t 6 n = ar, t 4 6 Br, t = 0 /Br, t /t k 2 J /Dr, t + 4^k J J r, t 4 H r , t = 2W 0 o k1
*Physics
411 EXAM COVER SHEET.*
You can write any formulas you like on the printed side of these sheets and bring them to the exam. _ Dr = PEr 4 6 Dr, t = 4^k 1 O o _ 0 r, t D 1 r, t ? D 2 r, t 6 n = ar, t 4 6 Br, t = 0 /Br, t /t J /Dr, t + 4^k 2 J J 0 r,
Syllabus for Physics 411: Electrodynamics I Text: Classical Electrodynamics, Third Edition, by J. D. Jackson, John Wiley & Sons, NY 1999 Electrostatics and Maxwells Equations: Coulombs law, Gauss law, point charges and the Dirac Delta function, surface ch