Question

Consider two conducting spheres of radill `R_(1) and R_(2) with R_(1) gt R_(2)`. If the two are at the same potential, the larger sphere has 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.

Answer 1

Here, `V_(1) = V_(2)`
`(Q_(1))/(4pi in_(0) R_(1)) = (Q_(2))/(4pi in_(0) R_(2)) :. (Q_(1))/(Q_(2)) = (R_(1))/(R_(2))` …(i)
As `R_(1) gt R_(2). :. Q_(1) gt Q_(2)`
i.e., larger sphere has more charge than the smaller sphere.
Now, `sigma_(1) = (Q_(1))/(4pi R_(1)^(2)) and sigma_(2) = (Q_(2))/(4pi in R_(2)^(2)) :. (sigma_(2))/(sigma_(1)) = (Q_(2))/(R_(2)^(2)) . (R_(1)^(2))/(Q_(1))`
Using (i) , `(sigma_(2))/(sigma_(1)) = (R_(2))/(R_(1)) . (R_(1)^(2))/(R_(2)^(2)) = (R_(1))/(R_(2))`
As `R_(1) gt R_(2)`, therefore, `sigma_(2) gt sigma_(1)`
i.e., charge density of smaller sphere si more than the charge density of larger sphere.