- #1
KvGroOve
Homework Statement
Taken from Purcell Problem 1.33
Consider the electric field of two protons a distance b apart. The potential energy of the system ought to be given by
U=∫E2dv.
Let E1 be the field of one particle alone and E2 that of the other. Evaluate
ε0∫E1⋅E2dv.
Set one of the protons at the origin and the other on the polar axis. Perform the integration over r before the integration over θ. Show that the integral has the value e2/4πε0b.
2. Homework Equations
The Attempt at a Solution
I uploaded my attempt. I started by applying Coulomb's Law for Electric Fields to both protons. I broke down r1 and r2 into their Cartesian components to perform the dot product between the two. Since the two charges were both on the Polar Axis (I think I chose the right axis), I set φ1=φ2. I assumed that r1=r and θ1=θ. I eventually got an expression involving r22 and Cos[θ2-θ]. I was able to rewrite r22 using the law of cosines. I have no idea how to address θ2. I figured that I could use the law of sines to figure it out but the expression turns out to be really messy. I'm not really sure if I'm on the right track or if I've made an error anywhere.
Any advice regarding my mistakes and/or pointing me in the right direction would be much appreciated! Thanks.