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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