课程名称︰电磁学上 课程性质︰必修 课程教师︰蔡尔成 开课学院：理学院 开课系所︰物理系 考试日期（年月日）︰103/11/11 考试时限（分钟）：110分钟 试题 : 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 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)Pl=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_l is 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 the 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? -- 感恩尔成! 赞叹尔成! --

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