Energy stored by a point charge

[kayethetutor]

Homework Statement

9uC charge at origin. How much energy does its electric field store outside a sphere centered about the origin with a radius of 5mm?

Homework Equations

C(sphere)=ab/(k(b-a)) V=kq/r U(cap)=(1/2)CV^2 E=kq/(r^2)
a=inner radius b=outer radius

The Attempt at a Solution

let the outer radius -> infinity, then C=a/k. V is 0 at infinity.
U=(1/2)CV^2=(1/2)(a/k)(kr/r)^2=(1/2)aqE= 72.9J which is the answer given in the book
This solution is something I dimly remember seeing but my reasoning is very shaky.
Why doesn't integrating ((1/2)εE^2)4∏r^2dr from r to infinity work? I get integral=-(1/2)ε(kq)^2(4∏/r) which is not at all the same

 

[mfb]

U= [...] = (1/2)aqE

E is the electric field at a radius of a? So ##E=\frac{k q}{ a^2}## and therefore
$$U=\frac{1}{2}aq \frac{k q}{ a^2} = \frac{1}{2} \frac{k q^2}{a}$$
Using ##k=\frac{1}{4\pi\epsilon}## in your equation gives the same formula.

 

[rude man]

1. Why doesn't integrating ((1/2)εE^2)4∏r^2dr from r to infinity work? I get integral=-(1/2)ε(kq)^2(4∏/r) which is not at all the same

It does! Did you run the numbers & not get the advertised answer?