Question

Two metal sphere of radius \(R\) and \(3R\) have same surface charge density \(σ\). If they are brought in contact and then separated, the surface charge density on smaller and bigger sphere becomes \(σ_1\) and \(σ_2\), respectively. The ratio \(\frac{σ_1}{σ_2}\) is.

• A. \(\frac{1}{9}\)
• B. \(9\)
• C. \(\frac{1}{3}\)
• D. \(3\)

Hint:

After contact both spheres reach the same potential, so $\sigma\propto\frac1R$.

Answer 1

The Correct Option is D

Step 1: Same potential
Spheres in contact share a potential: \(\frac{kq_1}{R}=\frac{kq_2}{3R}\), so \(q_2=3q_1\).

Step 2: Surface densities
\(\sigma_1=\frac{q_1}{4\pi R^2}\) and \(\sigma_2=\frac{q_2}{4\pi(3R)^2}=\frac{3q_1}{36\pi R^2}\).
\(\frac{\sigma_1}{\sigma_2}=\frac{q_1/(4\pi R^2)}{3q_1/(36\pi R^2)}=\frac{36}{12}=3\). Option (D).

Final Answer:
The ratio \(\frac{\sigma_1}{\sigma_2}=3\), option (D). \[ \boxed{\text{(D) }3} \]