A spherical shell is uniformly charged with the surface density `sigma`. Using the energy conservation law, find the magnitude of the electric force acting on a unit area of the shell.

Explanation

Energy of the chagred sphere of radius `r`, from the equation
`U = (1)/(2) q varphi = (1)/(2) q (q)/(4pi epsilon_(0) r) = (q^(2))/(8pi epsilon_(0) r)`
If the radius of the shell changes by `dr` then work done is
`4pi r^(2) F_(u) dr = -dU = q^(2)//8pi epsilon_(0) r^(2)`
Thus sought force per unit area,
`F_(u) = (q^(2))/(4pi r^(2) (8pi epsilon_(0) r^(2))) = ((4pi r^(2) sigma)^(2))/(4pi r^(2) xx 8pi epsilon_(0) r^(2)) = (sigma^(2))/(2 epsilon_(0))`