課程名稱︰電磁學
課程性質︰物理系必修
課程教師︰賀培銘
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2009/11/11
考試時限（分鐘）：180 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
電磁學-2009期中考 [若答案完全正確，沒有計算過程不扣分；但答案未化約到最簡形式會
酌量扣分]
1.(10%) Find A(s0,ψ0)defined by (所有s0>0,0≦ψ<2π)
δ(x-s0cosψ0)δ(y-s0sinψ0)=A(s0,ψ0)δ(s-s0)δ(ψ-ψ0), (1)
where x,y are 2D Cartesian coordinates, and s,ψ are 2D polar coordinates.
2.(10%)
∞
∫ dx e^(-ax^2/2) d2/dx2(δ(x^2-bx)) = ? (a,b屬於R) (2)
-∞
3.(10%) For a region V including the origin, evalute
^
∫ dτr /r^2‧▽f(r) (3)
V
where f(r) vanishes on the boundary of V
4.(25%) Consider a system of conductors as in Fig.1 The concentric conducting
shells have inner radii a1, a2 and outer radii b1, b2. The sphere of radius
a has charge Q. There is no net charge on the inner shell, and the outer
shell has total charge -Q. Find (a) the potential V(r) in the regin a<r<a1.
(b) the surface charge density σat r=b1. (c) the pressure on the surface at
r=a2. (d) the capacitance C od the system. (e) the total electrostatic
energy.
5.(10%) For the charge distribution ρ(r)=qδ^3(r)+kδ(r-R) for given constants
q,k, fin the total electrostatic energy.
6.(15%) Find the potential in the region 0<r<R for the boundary condition
V(R,θ)=kcos(2θ).
7.(20%)Consider a grounded spherical conducting shell with inner radius b and a
concentric thin insulator of radius a on which a surface charge density
σ(θ)=kcosθ is glued (see Fig.2). Find the potential V(r,θ) for both
region a>r and a<r<b
Derivation in spherical coordinates:
▽f
▽‧A
▽^2(f)
的極座標公式都有給
也有給
General solution of Laplace equation in spherical coordinates:
Fig.1:三個同心球殼 最內圈為a 第二圈為內徑a1 外徑b1 最外圈為 內徑a2 外徑b2
Fig.2:兩個同心球殼 內圈半徑a 外圈半徑b
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