Inclasswecalculatedtheelectrostaticself-energyofauniform-densitysphereofcharge,i.e.,theworkrequiredtoassemblethespherefrom"infinitedispersal."a.ALongthesamelines,calculatetheelectrostaticselfenergyofasphericalshellof

Question

In class we calculated the electrostatic self-energy of a uniform-density sphere of charge,i.e.,the work required to assemble the sphere from "infinite dispersal."
a.ALong the same lines,calculate the electrostatic self energy of a spherical shell of charge,of negligible thickness,radius a,and total charge Q.Note that the radius of the charge distribution does not vary as you build it up.
b.Calculate,explicitly,the total energy associated with the electric field of a thin spherical shell of charge,of radius a and charge Q,surrounded by vacuum.CAUTION:You may not simply assert that this is the same as the result of part a.

Answer 1

a.由自由能计算得W=Q^2/(8πεa)
b.由空间静电能密度1/2*εE^2计算
W = ε/2*∫Q^2/(πεr^2)^2*r^2 drdΩ
Ω积分域为全球面,r从a到∞
= Q^2/(8πε)*∫1/r^2 dr
=Q^2/(8πεa)
由计算结果两者相同