Two isolated metallic solid spheres of radii $R$ and 2R are charged such that both have same charge density $\sigma$. The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is $\sigma^{\prime}$. The ratio $\frac{\sigma^{\prime}}{\sigma}$ is :

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Electrostatics

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

\frac{σ ^{'}}{σ}

is :

Answer: (D)

Before contact each sphere carries

Q = σ \cdot 4 π r^{2}

, so

Q_{1} = 4 π R^{2} σ

and

Q_{2} = 16 π R^{2} σ

, a total of

20 π R^{2} σ

. Joining the spheres by a wire brings them to a common potential,

\frac{Q _{1}^{'}}{R} = \frac{Q _{2}^{'}}{2 R}

, i.e.

Q_{2}^{'} = 2 Q_{1}^{'}

, so charge conservation gives

Q_{2}^{'} = \frac{40 π R ^{2} σ}{3}

. The new density on the larger sphere is

σ^{'} = \frac{Q _{2}^{'}}{4 π ( 2 R ) ^{2}} = \frac{5 σ}{6}

, hence

\frac{σ ^{'}}{σ} = \frac{5}{6}

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Single correct option