State Gauss's theorem. Obtain an expression for elactric field at any point outside a charged spherical hollow conductor (shell).

Explanation

Statement: The total electric flux through a closed surface in free space is equal to `1//epsilon_(0)` times the net charge enclosed by the surface. Where `epsilon` is the absolute permitivity of free space.
Derivation :

In the above figure
E is the electric field.
R is the radius of a thin spherical shell of centre O.
r is the distance between the point P and O.
P is a point at distance r from O.
`DeltaS` is the area element around the point P.
Let q be the charge uniformly distributed over the surface of the shell. The electric flux through the Gaussian surface is given by
`phi = sum EDeltaS cos theta " " .......(1)`
The angle `theta` between E and `DeltaS` is `0 :. cos0 = 1`
` (1) implies phi sumEDetlaS`
`phi = EsumDeltaS" "....(2)`
where `sumDeltaS` = area of the spherical Gaussian surface `= 4pir^2`
`(2) implies phi = E xx 4pir^(2) " "...(3)`
From Gauss law, `phi= q/(epsilon_(0))" "...(4)`
On comparing the equations (3) and (4) we get,
`E xx4pir^(2) =q/(epsilon_(0))`
`E = q/(4piepsilon_(0)r^2)`
`E = 1/(4piepsilon_(0))q/r^2` Along OP produced