# Energy stored by a point charge

1. Mar 5, 2013

### kayethetutor

1. The problem statement, all variables and given/known data
9uC charge at origin. How much energy does its electric field store outside a sphere centered about the origin with a radius of 5mm?

2. Relevant equations
C(sphere)=ab/(k(b-a)) V=kq/r U(cap)=(1/2)CV^2 E=kq/(r^2)

3. 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

Last edited: Mar 5, 2013
2. Mar 5, 2013

### Staff: Mentor

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.

3. Mar 5, 2013

### rude man

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