Question

Two metallic shells \(A\) and \(B\) of radii \(3\,\text{cm}\) and \(4\,\text{cm}\) are given electric charges \(20\,\mu\text{C}\) and \(40\,\mu\text{C}\) respectively. If the shells are arranged concentrically, then the ratio of the surface charge densities of the shells \(A\) and \(B\) is

• A. \(4:27\)
• B. \(4:3\)
• C. \(16:9\)
• D. \(16:27\)

Hint:

For concentric conducting shells, - the induced charge on the inner surface of the outer shell equals the negative of the charge enclosed, - the remaining charge resides on the outer surface. Also, \[ \boxed{ \sigma=\frac{Q}{4\pi R^2}. } \]

Answer 1

The Correct Option is D

Step 1: Determine the final charge distribution. When the two conducting shells are arranged concentrically, - Shell \(A\) (inner shell) retains its charge \[ 20\,\mu\text{C} \] on its outer surface. - On shell \(B\), an induced charge \[ -20\,\mu\text{C} \] appears on its inner surface. Since the total charge on shell \(B\) is \[ 40\,\mu\text{C}, \] the charge on its outer surface becomes \[ 40-(-20)=60\,\mu\text{C}. \]

Step 2: Calculate the surface charge densities. For shell \(A\), \[ \sigma_A = \frac{20}{4\pi(3)^2}. \] For shell \(B\), \[ \sigma_B = \frac{60}{4\pi(4)^2}. \]

Step 3: Find the ratio. Therefore, \[ \frac{\sigma_A}{\sigma_B} = \frac{20}{36} \times \frac{64}{60} = \frac{16}{27}. \] Hence, \[ \boxed{\sigma_A:\sigma_B=16:27.} \] Therefore, the correct option is \(\boxed{(D)}\).