Question

Three concentric metallic spherical shells of radii \(R\), \(3R\) and \(9R\) are given charges \(Q_{1}\), \(Q_{2}\) and \(Q_{3}\), respectively. It is found that surface charge densities on the outer surfaces of the shells are equal. Then the ratio of charges given to shells \(Q_{1}:Q_{2}:Q_{3}\), is (Neglect the induction effects)

• A. \(1:4:16\)
• B. \(1:3:9\)
• C. \(\frac{1}{3}:3:27\)
• D. \(1:\frac{1}{3}:9\)

Hint:

Equal surface charge density means the charge on each shell is proportional to the square of its radius.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Induction is neglected, so each shell's charge sits on its own outer surface, with \(\sigma=\frac{Q}{4\pi r^{2}}\).

Step 2: Equal densities
\(\frac{Q_{1}}{R^{2}}=\frac{Q_{2}}{(3R)^{2}}=\frac{Q_{3}}{(9R)^{2}}\), so \(Q_{1}:Q_{2}:Q_{3}=1:9:81\).

Step 3: Scale the ratio
Dividing every term by 3 gives \(\frac{1}{3}:3:27\).

Step 4: Compare the options
This matches option (C).

Final Answer:
The charges are in the ratio \(\frac{1}{3}:3:27\). \[ \boxed{\text{(C) }\dfrac{1}{3}:3:27} \]