Pressure on uniformly charge spherical shell

[timhunderwood]

Homework Statement

First part of the problem was to work out E-field of uniformly charged spherical shell with charge Q and Radius R.

This was fine : E = 0 for r<R
and E = Q/(4*Pi*eps*r²) for r>R

QUESTION:
Find the pressure exerted on the shell due to the charges on its surface.

Homework Equations

The Attempt at a Solution

I tried to do it using the principle of virtual work, I get an answer and was hoping someone could confirm it is right?

I did this:
u_s, Energy stored in Electric field = \inteps*E²/2 (integrate over all space)

I did this in spherical polar coordinates and get U_s = Q²/(8*Pi*eps*R)

I then said (principle of virtual work:)

F, Force*dR = \partialU_s/\partialR *dR
and solved this to get:
F = -Q²/(8*Pi*eps*R²)

dividing by area of shell then gives

pressure = Q²/(32*Pi²*eps*R²)

is this right? If there's a mistake I think I may have not used the principle of virtual work correctly...

Thanks very much.

 

[jdwood983]

I do believe you are making this a little more complicated than necessary. The electrostatic pressure on the surface is given by the formula

P=\frac{\varepsilon_0}{2}E^2

(cf. Griffiths Equation 2.52)

 

[diazona]

pressure = Q²/(32*Pi²*eps*R²)

is this right? If there's a mistake I think I may have not used the principle of virtual work correctly...

You're missing a factor of R^2 in the denominator, but otherwise, yes, that is fine.

 

[timhunderwood]

Thats good,

I mistyped my last equation- it should have read R^4.

Thanks for the help