PHY 6346 Fall 2012 Homework #8, Due Monday, October 29 1. The general term in the Cartesian multipole expansion in (4.10) in the text is of the form l = Qij.k
(l) (l)
^ ^ ri rj . . . rk r l+1
(implicit summation over repeated indices), where the (constant
PHY 6346, Fall 2012 Homework #1, Due Monday, August 27 1. Suppose that, instead of scaling as 1/r 2 , the force between charges remains radial with linear superposition but varies as 1/r p for some p. Write an expression for the force F (magnitude and dir
PHY 6346 Fall 2012
Homework #3, Due Monday, September 10
1. A capacitor is made of two conducting electrodes bounded by surfaces S1 , S2 that carry
charges Q1 , Q2 in otherwise empty space.
(a) Show that the energy of charge Q2 in the potential produced b
PHY 6346, Fall 2013
Homework #1, Due Monday, August 26
1. Starting from equation (1.5) in the text, compute the electric eld () at every point in
space produced by a innite hollow cylindrical shell of radius b with uniform surface charge
density . The int
PHY 6346 Fall 2012 Homework #11, Due Wednesday, December 5 1. A time-independent electric charge distribution (x) and a current density J (x) produce an electrostatic field E = - , and a magnetic field B. (a) Show that the momentum contained in the combin
PHY 6346 Fall 2012 Homework #11, Due Monday, November 19 1. A magnetostatic field is due entirely to a localized permanent magnetization M0 . (a) Show that integrated over all space d3 x B H = 0.
(b) In Section 5.7 Jackson argues that the energy of a magn
PHY 6346 Fall 2012 Homework #10, Due Wednesday, November 14 1. Write E(-m) as an elliptic integral of positive argument. Write K(-m) as an elliptic integral of positive argument. What is the allowed range of negative argument? 2. A thin circular ring of r
PHY 6346 Fall 2012 Homework #9, Due Monday, November 5 We have to do one dielectric problem: 1. A point charge q is located in free space a distance d > a from the center of a dielectric sphere of radius a with permittivity = 0 . Both inside and outside t
PHY 6346 Fall 2013
Homework #2, Due Monday, September 2
1. What is x2 (x) ? What is x3 (x) ?
We understand -function expressions by how they behave in integrals against test functions.
Integrate by parts, twice:
dx f (x) x2 (x) = x2 f (x) (x)
+
(x2f ) (x)
PHY 6346 Fall 2013
Homework #3, Due Monday, September 9
1. A parallel plate capacitor consists of two large, at conducting plates of area A separated
by a small distance d. Equal and opposite charges Q, Q are placed on the two plates.
(a) What are the ele
PHY 6346 Fall 2013
Homework #11, Due Wednesday, December 4
1. A time-independent electric charge distribution () and a current density
an electrostatic eld = , and a magnetic eld
= .
() produce
(a) Show that the momentum contained in the combined electro
PHY 6346 Fall 2013
Homework #9, Due Monday, November 4
1. Show how to write E (m) as an elliptic integral of positive argument. Show how to write
K (m) as an elliptic integral of positive argument. What is the allowed range of positive
arguments? What is
PHY 6346 Fall 2013
Homework #8, Due Monday, October 28
1. The general ( ) term in the Cartesian multipole expansion in (4.10) in the text is of
the form
(l) xi xj . . . xk
l = Qij.k
r 2l+1
(l)
(implicit summation over repeated indices), where the (constan
PHY 6346 Fall 2013
Final Exam
Name:
This is a closed-book exam. Some problems ask for numerical answers, but only to order of
magnitude; calculators are unnecessary and not permitted.
Some possibly useful information:
Plm (x) = ( 1)m (1
m
( 1)m
2 m/2 d
(1
PHY 6346 Fall 2013
Homework #5, Due Monday, September 23
1. A long square cylinder of side a has potential = 0 at x = 0 and x = a, =
y = 1 a, and = +V0 at y = + 1 a.
2
2
V0 at
(a) What is (x, y ) in the cylinder interior? What is at the center? Sketch (y
PHY 6346 Fall 2013
Homework #6, Due Monday, September 30
1. A conducting cylinder of radius b is placed in a uniform electric eld
the resulting potential for > b?
The general solution to
= E0 . What is
2 = 0 in cylindrical coordinates is
=
(Am m + Bm m )
PHY 6346 Fall 2013
Homework #4, Due Monday, September 16
1. A potential (x, y, z ) satises 2 = 0 in the volume V = z
condition S = VS (x, y ) specied on the surface S = z = 0 .
0 with boundary
(a) Write the Greens function G(, ) within V and its normal de
PHY 6346 Fall 2012 Homework #7, Due Monday, October 8 1. Express (x) as a series 2. Suppose f (x) = an Pn (x).
1 an Pn (x). Express the integral -1 [f (x)]2 dx as a series of the an .
3. Express f (x) = |x| as a series
an Pn (x).
4. What is the value of t
PHY 6346 Fall 2012 Homework #6, Due Monday, October 1 1. A volume V is bounded by an outer cylindrical shell = b, where the potential is held fixed at +V0 0< |b = -V0 - < < 0, and an inner grounded, neutral conducting coaxial cylinder of radius = a. (a) F
PHY 6346 Fall 2012 Homework #8, Due Monday, October 29 1. The general term in the Cartesian multipole expansion in (4.10) in the text is of the form l = Qij.k
(l) (l)
^ ^ ri rj . . . rk r l+1
(implicit summation over repeated indices), where the (constant
PHY 6346 Fall 2012 Homework #6, Due Monday, October 1 1. A volume V is bounded by an outer cylindrical shell = b, where the potential is held fixed at +V0 0< |b = -V0 - < < 0, and an inner grounded, neutral conducting coaxial cylinder of radius = a. (a) F
PHY 6346 Fall 2012 Homework #5, Due Monday, September 24 1. The general solution to Laplace's equation in a rectangular box of dimensions a b c with only the top face z = c held at nonzero potential is
(x, y, z) =
n=1 m=1
Anm sin(
2 nm =
my nx ) sin( )
PHY 6346 Fall 2012 Homework #4, Due Monday, September 17 1. A potential (x, y, z) satisfies 2 = 0 in the volume V = cfw_z 0 with boundary condition |S = VS (x, y) specified on the surface S = cfw_z = 0. (a) Write the Green's function G(x, x ) within V and
PHY 6346 Fall 2012 Homework #3, Due Monday, September 10 1. A capacitor is made of two conducting electrodes bounded by surfaces S1 , S2 that carry charges Q1 , Q2 in otherwise empty space. (a) Show that the energy of charge Q2 in the potential produced b
PHY 6346 Fall 2012 Homework #2, Due Monday, September 3 1 . r Ignoring for the moment what happens at r = 0, the first gradient gives xj 1 = j (xk xk )-1/2 = - 1 (xk xk )-3/2 (2jk xk ) = - 3 , j 2 r r and the second leads to xj -3/2 = (x x )-3/2 + x - 3 (
PHY 6346 Fall 2012
Homework #5, Due Monday, September 24
1. The general solution to Laplaces equation in a rectangular box of dimensions a b c
with only the top face z = c held at nonzero potential is
(x, y, z ) =
Anm sin(
n=1 m=1
where
2
nm =
my
nx
) sin
PHY 6346 Fall 2012 Homework #9, Due Monday, November 5 We have to do one dielectric problem: 1. A point charge q is located in free space a distance d > a from the center of a dielectric sphere of radius a with permittivity = 0 . Both inside and outside t