Using the concept of energy density , show that the total energy stored by a shell of radius R and charge Q is `(Q^(2))/( 8 pi epsilon_(0)R)`

Explanation

The electric field field at a distance r from centre is
`E = (Q)/( 4pi epsilon_(0)r^(2))`
The energy stored in a shell of radius r and thickness dr
`dU = ( 1)/(2) epsilon_(0)E^(2) xx 4pi r^(2) dr`
`:. dU =(1)/(2)epsilon_(0)((Q)/(4pi epsilon_(0)r^(2)))^(2) xx 4p r^(2) dr ` `[ :' E = ( Q)/( 4pi epsilon_(0)r^(2))]`
or, `dU = ( Q^(2)) /( 8 pi epsilon_(0) )(dr)/(r^(2))`
`implies ` Total energy stored `U = ( Q^(2))/( 8 pi epsilon_(0)) int_(R)^(infty)(dr)/(r^(2))`
`implies U = ( Q^(2))/( 8 pi epsilon_(0))[ ( -1)/( r)]_(n)^(infty)= ( Q^(2))/( 8 pi epsilon_(0) R)`