Three concentric metallic spherical shells of radii \( R \), \( 2R \) and \( 3R \) are given charges \( Q_{1} \), \( Q_{2} \) and \( Q_{3} \), respectively. If the surface charge densities on the outer surfaces of these shells are equal, then, the proportion of charges \( Q_{1} : Q_{2} : Q_{3} \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 1 : 3 : 5 \)
Charge on a conductor resides on its surfaces, and induction makes the inner surface of each outer shell carry a charge equal and opposite to the total charge inside it. So the charge on the outer surface of each shell is the total charge enclosed by that surface.
Shell of radius \( R \): \( q_{1} = Q_{1} \)
Shell of radius \( 2R \): \( q_{2} = Q_{1} + Q_{2} \)
Shell of radius \( 3R \): \( q_{3} = Q_{1} + Q_{2} + Q_{3} \)
Equal surface charge densities \( \sigma = q/(4\pi r^{2}) \) mean each outer-surface charge is proportional to \( r^{2} \):
\( q_{1} : q_{2} : q_{3} = R^{2} : (2R)^{2} : (3R)^{2} = 1 : 4 : 9 \)
Take \( q_{1} = 1 \) unit:
\( Q_{1} = 1 \)
\( Q_{2} = 4 - 1 = 3 \)
\( Q_{3} = 9 - 4 = 5 \)
\( Q_{1} : Q_{2} : Q_{3} = 1 : 3 : 5 \)
Note: A similar question was also asked in CUET (UG) 2025 on 30-05-2025 (Slot 2).