PHYS809 Class 7 Notes
The method of images
The method of images is useful for finding the potential of a
charge distribution in the presence of grounded conductors of
simple geometry. The simplest case is when the surface of the
conductor is an infinite
Homework 3 solutions
1.3 (a) The charge density will be of form k ( r R ) , where k is a constant. Integrating over a
spherical volume with radius greater than R, we find that k = Q ( 4 R 2 ) . Hence
(x) =
Q
(r R).
4 R 2
(b) Similarly to part (a) by int
PHYS809 Class 35 Notes
Energy density and energy flux
In looking for solutions to Maxwells equation, we found it convenient to consider physical quantities as
the real parts of complex expressions, e.g.
i k x t )
,
E ( x, t ) = Re E0 e (
(35.1)
where the
PHYS809 Class 33 Notes
Refraction and reflection of electromagnetic plane waves
Suppose two linear media are separated by a plane. A monochromatic wave incident on the plane in
medium 1 will, in general, result in a reflected wave in medium 1 and a trans
PHYS809 Class 32 Notes
Electromagnetic plane waves
Maxwells equations for uniform linear media with no sources are
E
= 0,
t
B
E +
= 0.
t
E = 0, B
B = 0,
(32.1)
With the understanding that physical quantities are given by taking the real part of comple
PHYS809 Class 31 Notes
Momentum and the electromagnetic stress tensor
We now consider conservation of momentum. Let Ppart be the total momentum of all the particles in a
fixed volume V. Using the Lorentz force on a single particle of charge q,
F = q (E +
PHYS809 Class 30 Notes
Conservation of energy for electromagnetic fields
We begin by considering electromagnetic waves propagating in vacuum. The effects of linear media will
be considered later. At a fundamental level the waves are due to motion of char
PHYS809 Class 29 Notes
Maxwells equations and electromagnetic waves
In complete vacuum (i.e. no sources present), Maxwells equations are
E = 0,
1
0
B 0
E
= 0,
t
B
B = 0, E +
= 0.
t
(29.1)
Taking the curl of the equations containing time derivatives an
PHYS809 Class 28 Notes
Time varying fields and Maxwells equations
Faraday experimentally determined that a transient current is induced in a circuit if (a) a steady current
in a nearby circuit is switched on or off, (b) a nearby circuit with a steady cur
PHYS809 Class 27 Notes
Boundary value problems in magnetostatics
A variety of techniques are available for solving boundary value problems in magnetostatics. For linear
media with piecewise uniform magnetic permeabilities, the vector potential in the Cou
Homework 4 solutions
2.14 (a) Noting that the boundary conditions are an odd function of , the series solution for the
potential inside the cylinder is of form
m
( , ) = am sin m .
b
m =1
The coefficients are
am =
1
2
( b, ) sin m d =
0
1
2
V sin m d
Homework 5 solutions
3.2 First consider the related problem of the potential due to a spherical cap with charge density
Q ( 4 R 2 ) . The potential on the symmetry axis is given by the integral
1
Q
Q
2 R 2 sin d
4 0 4 R 2 ( R + z 2 2 Rz cos )1 2
0
( z) =
Homework 6 solutions
3.23. The potential is the solution of
1 1 2 2
q ( ) ( ) ( z z )
.
+2
+ 2 =
2 z
0
To obtain each of the first two forms of the potential, we make use of the completeness relation
( ) =
1
2
e
im( )
.
m =
Let
( , , z ) =
1
2
fm ( ,
November 9th 11th, 2011
PHYS809 Fall 2011 Second Take-home Exam
Please write your name on each answer sheet.
Since this is a take home exam, there are a few rules that you must follow. You cannot discuss or communicate
about the exam with any one by any m
October 12th 14th, 2011
PHYS809 Fall 2011 First take-home exam
Please write your name on each answer sheet.
Since this is a take home exam, there are a few rules that you must follow. You cannot discuss or
communicate about the exam with any one by any me
USEFUL FACTS AND FORMULAE FOR
LEGENDRE POLYNOMIALS
1. The differential equation
The Legendre polynomials Pl(x), l = 0, 1, . are a set of orthogonal polynomials over the range x
[-1,1]. Pl(x) is of degree l and is a solution of the differential equation
d
Homework 2 solutions
2.3 Find the potential energy of a uniformly charged sphere of charge Q and radius R by integrating
(a) ,
(b) E2.
The charge density insider the sphere is = 3Q 4 R 3 . Hence the electric field is
Qr R3 , r < R,
E=
2
Q / r , r R.
The
Homework 2 solutions
1. Find the potential energy of a uniformly charged sphere of charge Q and radius R by integrating
(a) ,
(b) E2.
2. Find the dipole moment of :
(a) a straight wire of length L with a linear charge density
(z) =
qz
,
L2
z < L 2,
(b) a
Homework 1
1. The Maxwell relationship between a time-independent magnetic field B and the current
density J producing it is B = 0 J. A long cylinder of conducting ionized gas (plasma)
occupies the region < a (in cylindrical polar co-ordinates).
(a) Show
Homework 1
1. The Maxwell relationship between a time-independent magnetic field B and the current
density J producing it is B = 0 J. A long cylinder of conducting ionized gas (plasma)
occupies the region < a (in cylindrical polar co-ordinates.
(a) Show t
Homework 8 solutions
5.20. (a) The force is
F = ( M ) Bd 3 x + ( M n ) Bda.
V
S
Consider the ith component of the first integrand. This is
M m
M m
M i
M k
( M ) B i = ijk jlm
Bk = ( kl im km il )
Bk =
Bk
Bk
xl
xl
xk
xi
= Bk
M i ( Bk M k )
B
+ Mk k
xk
xi
Homework 7 solutions
5.3. Consider a single current loop in the x-y plane. On the axis, the magnetic scalar potential for
the magnetic induction is
(z) =
0 I z
2 z
.
2
2 12
z +a
z
(
)
Hence the magnetic induction on the axis is
Bz ( z ) =
0 I
1
2 z 2 + a
PHYS809 Class 26 Notes
Induced and permanent magnetization terminology
Due to the motion of their electrons, some atoms and molecules can have intrinsic magnetic moments.
Other atoms develop magnetic moments when placed in an external magnetic field. If
PHYS809 Class 25 Notes
Magnetic dipole moment
The dipole part of the scalar potential for a current loop can be written in the form
md =
0 m r
,
4 r 3
( r > 0) ,
(25.1)
where for a circular loop of radius a and unit normal k,
m = a 2 Ik.
(25.2)
(Note th
PHYS809 Class 10 Notes
Separation of variables in Cartesian geometry
Separation of variables in Cartesian geometry, with ( x, y, z ) = X ( x ) Y ( y ) Z ( z ) , leads to the set of
equations
d2X
=X,
dx 2
d 2Y
= Y ,
dy 2
d 2Z
= Z,
dz 2
(1.1)
+ + = 0.
(1.
PHYS809 Class 9 Notes
Separation of variables in polar coordinates
To illustrate the method of separation of variables, consider the two
dimensional problem in which two semi-infinite conducting planes at
potential V meet along their edges at an angle .
PHYS809 Class 8 Notes
The Dirichlet Green function for a sphere
The Dirichlet Green function for a sphere is simply the solution found earlier for a point charge in the
presence of a grounded sphere. For the exterior problem, unit charge is placed at x o
PHYS809 Class 6 Notes
Green function methods in electrostatics
Greens first theorem is
+ dV = dS = n dS ,
2
V
S
(1.1)
S
where and are two scalar fields and S is the bounding surface of a volume V. Also the unit vector n
is normal to the surface S and po
PHYS809 Class 5 Notes
Conductors
In a conductor, there are free charges that move in the presence of an electric field. A consequence is
that in a purely electrostatic situation, the electric field inside a conductor must be zero. Since the
electric fiel
PHYS809 Class 4 Notes
Electrostatic energy
The electrostatic energy of a pair of charges q1 and q2 at positions r1 and r2 is
U=
1
q1q2
= q21 ( r2 r1 ) = q12 ( r1 r2 ) ,
4 0 r1 r2
where 1 ( r ) is the potential due to charge q1.
The electrostatic energy o