Two isolated metallic solid spheres of radii R and 2R are charged such that both of these have same charge density `sigma`. The spheres are located far away from each other and connected by a thin conducting wire. Find the new charge density on the bigger sphere.

Answer

`00.83`

Explanation

Radius of the smaller sphere is R.
`:. q_1= sigma (4piR^2)`
Similarly, for bigger sphere,
`q_2=sigma(4pi)(2R)^2=16sigmaR^2`
Total charge (q) on both the sphere is,
`q=q_1+q_2=20sigmapiR^2`
As, capacitance for spherical capacitors,C=`4piverepsilon_0R`
`:. C_1/C_2=R/(2R)`
`:. C_1/C_2=1/2`
Now after connecting, charge is distributed in the ratio of their capacities.
`:.q_1^./q_2^.=1/2`
`:.(q_1^.+q_2^.)/q_2^.=(1+2)/2=3/2`
As total charge is conserved,
`:. q_2^.=(2q)/3=(40)/3sigmapiR^2`
Now,
`sigma^.=(q_2^.)/(4pi(2R)^2)=(40/3sigmaR^2)/(16 pi R^2)=5/6 sigma`
`sigma^./sigma=5/6=0.83`