課程名稱︰電磁學上
課程性質︰物理系雙班必帶
課程教師︰王立民
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰99/12/16
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
考試範圍：CH3.2.2~CH4.3
1. Find the potential inside and outside a spherical shell that carries a
uniform surface charge σ0. (10%)
2. Using general solution with cylindrical symmetry:
∞
V(s,ψ)=a0+b0lns+Σ[s^k(akcoskψ+bksinkψ)+s^-k(ckcoskψ+dksinkψ)]
k=1
Find (a) the potential outside an infinitely long metel pipe, of radius R,
placed at right angles to an otherwise uniform electric field E0, and
(b) the surface charge induced on the pipe.(20%)
3. (a) Using the method of images, determine the electric field outside a
spherical conducting shell (radius R) placed in a uniform electric field E0.
[Hint: the same image charges as those for a conducting sphere, and using
1/r+ - 1/r- ≒ d/(r^2 cosθ), where d is the separated distance between r+
and r-.](10%)
(b) Find the induced surface charge density.(5%)
(c) Calculate the dipole moment due yo the induced surface charge.(5%)
^
4. A sphere of radius R carries a polarization P(r)= kr^2 r, where k is a
^
constant and r r is the vector from the center.
(a) Calculate the bound charge σb and ρb.(10%)
(b) Find the field inside and outside the sphere.(10%)
(c) Find the electric displacement inside and outside the sphere.(10%)
5. The electron cloud for a hydrogen atom in the ground state has a charge
density: ρ(r)=(q/πa^3)*e^(-2r/a), where q is the charge of the electron
and a is the Bohr radius.
(a) Find the atomic polarizability of this atom.(10%)
(b) A point charge Q is situated a large distance r(r>>a) from the neutral
hydrogen atom. Find the force of attraction between them.(10%)
Hint: using these formulas:
r
∫[e^(-2r/a)*r^2]dr=-0.5a[e^(-2r/a)*(r^2-ar+0.5a^2)-0.5a^2]
0
e^x = 1 + x + x^2/2! + x^3/3! +.....
^ ︿
The field due to a dipole: E(r,θ)=p/(4πε0r^3)*(2cosθ r+sinθ θ)
6. A very long cylinder, of radius a, carries a uniform (not "radial")
polarization P perpendicular to its axis, and there is no free charge on it.
(a) Find the electric field inside and outside the cylinder.(10%)
(b) Find the electric displacement inside and outside the cylinder.
Is it zero? And explain your answer.(10%)
Hint: using the approximation: (s and d are vectors.)
s-d/2 1 (s．d) d
──── ≒──(s+s───-─)
(s-d/2)^2 s^2 s^2 2
s+d/2 1 (s．d) d
─────≒──(s-s───+─), for |s|>>|d|
(s+d/2)^2 s^2 s^2 2
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