Sphere 1 with radius R_1 has positive charge q, Sphere 2 with radius 4.50 R_1 is far from sphere 1 and initially uncharged. The separated spheres are then connected with a wire then uncouth to retain only negligible charge. (a) What is the ratio V_1/V_2 of the final potentials of the spheres? (b) what fraction of q ends up on sphere? (c) What fraction of q ends up on sphere 2? (d) What is the ratio q_1/q_2 of the surface charge densities of the spheres?

capacitance of sphere 2 will be 4.5 times sphere 1

a) when spheres are in contact they will have same potential finally. So

V_1 / V_2 = 1

b)

Charge will be distributed in the ratio of their capacity

charge on sphere1 = q x 1 / (1 + 4.5)

= q / 5.5

fraction = 1 / 5.5

c) charge on sphere 2

= q x 4.5 / 5.5

fraction = 4.5 / 5.5

d) surface charge density of sphere 1

= q / (5.5 x A) where A is surface area

surface charge density of sphere 2

= q x 4.5 / (5.5 x 4.5² A) where A is surface area

= q / (5.5 x 4.5 A)

q_1/q_2 = 4.5