Question

Two isolated metallic solid spheres of radii R and 2R are charged such that both have same charge density σ . The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is σ' . The ratio σ'/σ is

(1) 9/4

(2) 4/3

(3) 5/3

(4) 5/6

Answer 1

Correct option is (4) 5/6

C₁ = 4πε₀R

C₂ = 4πε₀2R

Common potential

\(=\frac{c_1v_1+c_2v_2}{c_1+c_2}\)

\(=\frac{c_1+θ_2}{c_1+c_2}\)

\(=\frac{4\pi R^2σ+4\pi (2R)^2σ}{4\piε_0R+4\piε_0(2R)}\)

\(=\frac{5σR}{3ε_0}\)

So new charge on bigger sphere

θ = c₂v

\(=4\piε_0(2R)\times\frac{5σR}{3ε_0}\)

\(θ_2=\frac{40\pi R^2σ}{3}\)

New charge density on bigger one = \(\frac{40\pi R^2σ}{\frac{3}{A_2}}\)

\(σ^1=\frac{4\pi R^2σ}{3}\times\frac{1}{4\pi}(2R)^2\)

\(σ^1=\frac{5}{6}σ\)

\(\frac{σ^1}{σ}=\frac{5}{6}\)