Consider two conducting spheres of radii `R_(1)` and `R_(2)` with `R_(1) gt R_(2)`. If the two are at the same potential, the larger sphere bas more charge than the smaller sphere. State whether the charge density of the smaller sphere is more or less than that of the larger one.

Explanation

Both spheres have same potential
`:. V_(1) = V_(2)`
`:. (kq_(1))/(R_(1))=(kq_(2))/(R_(2))`
`:. (q_(1)R_(1))/(4piR_(1)^(2))= (q_(2)R_(2))/(4piR_(2)^(2))`
(` because ` Dividing by 4 pi on both sides )
`sigma_(1) R_(1) = sigma_(2) R_(2) [ because sigma= (q)/(4piR^(2))]`
Now `R_(1) gt R_(2)`
`sigm_(1) lt sigma_(2)`
Second Method :
Charge on larger sphere is more than charge on smaller sphere
Now `(kq_(1))/(R_(1)) = (kq_(2))/(R_(2))`
`[becauseq = sigmaA] `
and `k = (1)/(4piin_(0))`
`(sigma_(1)A_(1))/(4piin_(0)R_(1))=(sigma_(2)A_(2))/(4piin_(0)R_(2))`
`:. (sigma_(1)xx4piR_(1)^(2))/(4piin_(0)R_(1))=(sigma_(2)xx4piR_(2)^(2))/(4piin_(0)R_(2))`
`:. (sigma_(1)R_(1))/(in_(0))= (sigma_(2)R_(2))/(in_(0))`
`:. (sigma_(1))/(sigma_(2))=(R_(2))/(R_(1))`
but `R_(1) gt R_(2) implies 1 gt (R_(2))/(R_(1))`
`(sigma_(1))/(sigma_(2)) lt 1`
`:. sigma_(1) lt sigma_(2)`
`:.` Charge density of smaller sphere is less than that of larger sphere .