F07 Practice Midterm
2
Physics 505 Fall 2007 Practice Midterm The midterm will be a two hour open book, open notes exam. Do all three problems. 1. A rectangular box has sides of lengths a, b and c c x y z a b a) For the Dirichlet problem in the interior of the box, the Green’s function may be expanded as G(x, y, z ; x ,y ,z )= ∞ m=1 ∞ n=1 g mn (z,z ) sin mπx a sin mπx a sin nπy b sin nπy b Write down the appropriate differential equation that g mn (z,z ) must satisfy. b) Solve the Green’s function equation for g mn (z,z ) subject to Dirichlet boundary conditions and write down the result for G(x, y, z ; x ,y ,z ). c) Consider the boundary value problem where the potential on top of the box is Φ(x, y, c)= V (x, y) while the potential on the other five sides vanish. Using the Greens’ function obtained above, show that the potential may be written as Φ(x, y, z )= ∞ m=1 ∞ n=1 A mn sin mπx a sin nπy b sinh γ mn z where γ mn = π (m/a) 2 +(n/b) 2 and A mn = 4 ab sinh γ mn c a 0 dx b 0 dy V (x, y) sin mπx a sin nπy b Over -→
-
Upload
ankara271828 -
Category
Documents
-
view
216 -
download
5
description
e
Transcript of F07 Practice Midterm
-
Physics 505 Fall 2007
Practice Midterm
The midterm will be a two hour open book, open notes exam. Do all three problems.
1. A rectangular box has sides of lengths a, b and c
c
x
y
z
a
b
a) For the Dirichlet problem in the interior of the box, the Greens function may beexpanded as
G(x, y, z;x, y, z) =
m=1
n=1
gmn(z, z) sinmpix
asin
mpix
asin
npiy
bsin
npiy
b
Write down the appropriate differential equation that gmn(z, z) must satisfy.
b) Solve the Greens function equation for gmn(z, z) subject to Dirichlet boundaryconditions and write down the result for G(x, y, z;x, y, z).
c) Consider the boundary value problem where the potential on top of the box is(x, y, c) = V (x, y) while the potential on the other five sides vanish. Using theGreens function obtained above, show that the potential may be written as
(x, y, z) =
m=1
n=1
Amn sinmpix
asin
npiy
bsinh mnz
where mn = pi(m/a)2 + (n/b)2 and
Amn =4
ab sinh mnc
a0
dx
b0
dy V (x, y) sinmpix
asin
npiy
b
Over
-
2. The potential on the surface of a sphere of radius a is specified by
V0
0
0
V (, ) =
{ 0, 0 < V0, pi 0, pi < pi
There are no other charges in this problem.
a) Show that the potential outside the sphere may be expressed as
(r, , ) =
l=0,2,4,6,...
V0[Pl+1(cos) Pl1(cos)](ar
)l+1Pl(cos )
where we take P1(x) = 0. Note that Legendre polynomials satisfy the relation(2l + 1)Pl(x) = P l+1(x) P l1(x).
b) For fixed V0, what angle maximizes the quadrupole moment?
3. A line charge on the z axis extends from z = a to z = +a and has linear chargedensity varying as
x
0
0
|z |
z
a
a
y (z) ={0z
, 0 < z a0|z|, a z < 0
where is a positive constant. The total charge on the 0 < z a segment is Q (andthe charge on the a z < 0 segment is Q).a) Calculate all of the multipole moments of the charge distribution. Make sure to
indicate which moments are non-vanishing.
b) Write down the multipole expansion for the potential in explicit form up to thefirst two non-vanishing terms.
c) What is the dipole moment ~p in terms of Q, a and ?
