課程名稱︰電磁學上
課程性質︰物理系大二必修
課程教師︰蔡爾成
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2014/11/11
考試時限（分鐘）：110分鐘
試題 :
Introduction to E1ectrodynamis 11-11-2014
1. For the spherical polar coordinates,
^ ^ ^ ^ ^ ^
(a) [5%] express the unit vectors r, θ, ψ in terms of x, y, z.
(b) [5%] Also work out the inverse formula,
^ ^ ^ ^ ^ ^
giving x, y, z in terms of r, θ, ψ.
→ 3 ^
2. Suppose the electric field in some region is found to be E = kr r, in
spherical coordinates (k is some constant).
(a) [5%] Find the charge density ρ.
(b) [5%] Find the total charge contained in a sphere of radius R, centered at
the origin.
(Do it two different ways.)
3. [15%] A charge Q is distributed uniformly along the z axis from z = -a to
→
z = +a. Show that the electric potential at a point x is given by
Q 1 1 a 2 1 a 4
V(r,θ)=————— —— [ 1+——(——) P (cosθ)+——(——) P (cosθ)+...]
4πε0 r 3 r 2 5 r 4
⑨
2 3
1 t t
for r>a. [Hint] ln———= t+—— +——+...
1-t 2 3
4. [15%] How would you define the octopole moment? Express the octopole term
in the multipole expansion in terms of the octopole moment.
5. [10%] A sphere centered at the origin with radius R and charge Q
(a) distributed in the sphere with spherically symmetric volume charge density
ρ(r), (note that ρ(r) may not be uniform.)
or (b) evenly distributed over the spherical surface with surface charge
density σ = Q/4πR^2. Which configuration has the higher dcctrstatic
energy ? Why?
6. The Legendre Ploynomial P_l(x) is the solution of the ordinary Legendre
differential equation
d 2 dP_l
—[(1-x )———]+ l(l+1)P_l=0
dx dx
(a) [10%] Prove that
1
∫ P '(x)P (x)dx=0 if l≠l'.
-1 l l
(b) [5%] Given that
l
1 ∞ r< ^ ^
————=Σ ——— P_l(x‧x'),
→ → l=0 l+1
|x - x'| r>
Show that
1 ∞ l
————— = Σ t P (x).
2 l=0 l
√(1+t +2tx)
(c) [10%] Show that the normalization of P is
l
2
<P |P >=———
l l 2l+1
[Hint]
2
dx ln(1+t -2tx)
∫—————=-———————
2 2t
1+t +2tx
2 3
1 t t
ln———=t+——+——+...
1-t 2 3
7. A point charge q is placed a distance d>R from the center of an equally
charged isolated conducting sphere of radius R,
(a) [10%] What is the force exerted by the conducting sphere on the charge q?
(b) [5%] Show that the limiting value of tne force of attraction is
2
-q ^
————— r
2
16πε a
0
when the point charge is located at a distance a (=d-R) from the
surface of the sphere, if a<<R?