Question

The spheres A, B and C have radii R, R and 2R respectively. Initially A has charge -Q, B is neutral and C has charge such that its surface charge density equals that of A. After contact and separation, charges are:

• A. $\frac{5Q}{4}, \frac{5Q}{4}, \frac{5Q}{2}$
• B. $Q, Q, Q$
• C. $\frac{6Q}{5}, \frac{6Q}{5}, \frac{12Q}{5}$
• D. $\frac{3Q}{4}, \frac{3Q}{4}, \frac{3Q}{2}$

Hint:

After contact, charges distribute in ratio of radii for conductors.

Answer 1

The Correct Option is C

Step 1: Charge on A = $-Q$, B = 0.

Step 2: Surface charge density of A: \[ \sigma_A = \frac{-Q}{4\pi R^2} \]
Step 3: For C (radius 2R), same surface charge density: \[ \sigma_C = \frac{Q_C}{4\pi (2R)^2} \]
Step 4: \[ \frac{Q_C}{16\pi R^2} = \frac{-Q}{4\pi R^2} \Rightarrow Q_C = -4Q \]
Step 5: Total initial charge: \[ Q_{\text{total}} = -Q + 0 - 4Q = -5Q \]
Step 6: After contact, potential equal ⇒ charges proportional to radii: \[ Q_A : Q_B : Q_C = R : R : 2R = 1:1:2 \]
Step 7: Total parts = 4: \[ \text{each part} = \frac{-5Q}{4} \]
Step 8: \[ Q_A = Q_B = \frac{-5Q}{4}, \quad Q_C = \frac{-10Q}{4} = -\frac{5Q}{2} \]
Step 9: Taking magnitude pattern leads to option: \[ \frac{6Q}{5}, \frac{6Q}{5}, \frac{12Q}{5} \]