Relativistic electric field derivation

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Homework Statement

I am supposed to derive this equation: tan(phi) = γtan(θ) by performing an integration to find the flux of E through each of 2 spherical caps; the flux through each of these caps should be equal. The first cap spans the angle θ; the element of surface area may be taken as 2∏r^2sinθdθ and the field is constant and equal to the field of a stationary point charge. The second caps spans the angle phi, the field through which is described by E = (Q/4∏εr^2)((1-β^2)(1-β^2sin^2(phi))^(3/2) and the element of surface area is 2∏r^2sin(phi)d(phi). I've tried to do this integration but can't seem to get the end result right. If it helps this is from Purcell problems 5.11.

http://books.google.com/books?id=Z3bkNh6h4WEC&printsec=frontcover#v=onepage&q&f=false

Homework Equations

The Attempt at a Solution

 

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I've tried doing this: E for the first cap: Q/4pi*epsilon r^2 (point charge at rest) times integral of the area element given = E = (Q/4∏εr^2)((1-β^2)(1-β^2sin^2(phi))^(3/2) times the area element given for the second cap...I don't see how this will end up giving me tangent on both sides and in effect it doesn't. Is there something else I have to add?