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Question

A conducting sphere of radius 10 cm is charged with 10 C. Another uncharged sphere of radius 20 cm is allowed to touch it for some time. After that if the spheres are separated, then surface density of charges on the spheres will be in the ratio of

A
1:4
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B
1:3
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C
2:1
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D
1:1
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Solution

The correct option is C 2:1
When two conducting spheres touch each other, there will be a flow of charge until they both have the same potential.
Let, R1 and R2 be the radii of spheres 1 and 2 respectively. Let, Q1 and Q2 be the charges on sphere 1 and 2 respectively, after they are seperated since, both have same potential.
V=Q1C1=Q2C2
Q14πϵ0R1=Q24πϵ0R2
Q1Q2=R1R2
Surface charge density on sphere 1 is
σ1=Q14πR21
Surface charge density on sphere 2 is
σ2=Q24πR22
σ1σ2=Q14πR21×4πR22Q2=Q1Q2×R22R21
σ1σ2=R1R2×R22R21
σ1σ2=R2R1=2010
σ1σ2=21

σ1:σ2=2:1
(C).

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