Electrostatic force between 2 hemispheres

[mathnerd15]

Homework Statement

A metal sphere of radius R carries a total charge Q, what is the force of repulsion between the northern and southern hemispheres

Homework Equations

$$\large f=\sigma Eav=1/2 \sigma(Eabove+Ebelow). \\Fz=\int fz dA =\int \sigma Eav cos\theta R^{2}sin\theta d\theta d\phi=\int \frac{Q}{2*4\pi R^2}\frac{Q}{4\pi\epsilon o R^2}cos\theta R^{2}sin \theta d\theta d\phi$$

The Attempt at a Solution

I'm not sure why you don't integrate the R^2 in spherical coordinates? when you integrate are you summing force differential elements from each piece from the volume of the sphere? also since E is 0 inside the conductor and there is a force only outside, the force tends to separate the hemispheres and then doesn't the force cancel (this seems to be the calculation for one hemisphere only)? maybe I don't understand the integral sum of the force elements?

 

[vela]

You're not integrating over the volume; you're integrating over the area. A sphere is a two-dimensional surface.

 

[mathnerd15]

thanks, sorry so I am summing the differential force elements in the z direction from the surface of the sphere by taking Ez? but aren't there radial force elements in all directions which cancel in the +z -z with azimuthal symmetry? or if you calculate the force of each hemisphere then wouldn't the integration be from 0 to pi for phi instead of 2 pi?

do most people take mechanics before EM like from some text?