Question

Two isolated metallic solid spheres of radii $R$ and $2 R$ 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 :

• A. $\frac{5}{3}$
• B. $\frac{9}{4}$
• C. $\frac{5}{6}$
• D. $\frac{4}{3}$

Answer 1

The Correct Option is C

To solve this problem, we need to understand the concept of charge distribution when two metallic spheres are connected by a thin conducting wire.

Initially, we have two metallic spheres with radii \(R\) and \(2R\), both charged with the same charge density \(\sigma\). The charges on these spheres can be calculated as follows:

• The charge on the smaller sphere, \(Q_1 = \sigma \cdot 4\pi R^2\).
• The charge on the larger sphere, \(Q_2 = \sigma \cdot 4\pi (2R)^2 = \sigma \cdot 16\pi R^2\).

When the spheres are connected by a conducting wire, charge will redistribute until both spheres have the same potential. If \(q_1\) and \(q_2\) are the new charges on the smaller and larger spheres respectively, then:

• Total initial charge \(Q = Q_1 + Q_2 = \sigma \cdot 4\pi R^2 + \sigma \cdot 16\pi R^2 = \sigma \cdot 20\pi R^2\).
• Since the potential on both spheres must be the same, the potential of the spheres given by \(\frac{q_i}{r_i}\) will equalize:
• \(\frac{q_1}{R} = \frac{q_2}{2R} \Rightarrow q_2 = 2q_1\).

Using conservation of charge:

• \(q_1 + q_2 = Q = \sigma \cdot 20\pi R^2\).
• Substitute \(q_2 = 2q_1\) into the equation above:
• \(q_1 + 2q_1 = \sigma \cdot 20\pi R^2 \Rightarrow 3q_1 = \sigma \cdot 20\pi R^2 \Rightarrow q_1 = \frac{\sigma \cdot 20\pi R^2}{3}\).
• Thus, \(q_2 = 2 \cdot \frac{\sigma \cdot 20\pi R^2}{3} = \frac{40\pi R^2 \sigma}{3}\).

The new charge density \(\sigma'\) of the bigger sphere is:

• \(\sigma' \cdot 16\pi R^2 = \frac{40\pi R^2 \sigma}{3}\).
• Solving for \(\sigma'\):
• \(\sigma' = \frac{\frac{40\pi R^2 \sigma}{3}}{16\pi R^2} = \frac{40}{48} \sigma = \frac{5}{6} \sigma\).

Therefore, the ratio \(\frac{\sigma'}{\sigma}\) is \(\frac{5}{6}\).

Hence, the correct answer is

\(\frac{5}{6}\)