Three concentric metallic spherical shells of radii R, 2 R, and 3 R are given charges \( Q_1, Q_2 \) and \( Q_3 \), respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells \( Q_1 : Q_2 : Q_3 \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1 : 3 : 5
By electrostatic induction, the inner surface of each shell carries a charge equal and opposite to the charge enclosed, so the outer surface of each shell carries the total charge enclosed by it.
\( \text{outer surface of shell 1}: Q_1 \)
\( \text{outer surface of shell 2}: Q_1 + Q_2 \)
\( \text{outer surface of shell 3}: Q_1 + Q_2 + Q_3 \)
Equal surface charge densities \( \sigma = q/4\pi r^{2} \) give
\( \displaystyle \frac{Q_1}{4\pi R^{2}} = \frac{Q_1 + Q_2}{4\pi (2R)^{2}} = \frac{Q_1 + Q_2 + Q_3}{4\pi (3R)^{2}} \)
\( Q_1 + Q_2 = 4Q_1 \;\Rightarrow\; Q_2 = 3Q_1 \)
\( Q_1 + Q_2 + Q_3 = 9Q_1 \;\Rightarrow\; Q_3 = 5Q_1 \)
\( \displaystyle Q_1 : Q_2 : Q_3 = 1 : 3 : 5 \)
Note: A similar question was also asked in CUET (UG) 2026 on 24-05-2026 (Slot 1).