First slide

Redistribution of charges and concept of common potential

Question

A conducting body 1 has some initial charge Q, and its capacitance is C. There are two other conducting bodies, 2 and 3, having capacitance: C₂ = 2C and C₃ = $\infty$ . Bodies 2 and 3 are initially uncharged. Body 2 is touched with body 1. Then, body 2 is removed from body 1 and touched with body 3, and then removed. This process is repeated for N times. If the charge on body 1 at the end is found to be $\frac{Q}{x^{N}}$ , find the value of x.

Moderate

Solution

Initial charge on 1 = Q when C₁ and C₂ are touched
$\Rightarrow \frac{Q_{1}}{Q_{2}} = \frac{C}{2 C} = \frac{1}{2} \Rightarrow Q_{1} = \frac{Q}{3}, Q_{2} = \frac{2 Q}{3}$
Now when C₂ and C₃ are touched
$\Rightarrow \frac{Q_{2}}{Q_{3}} = \frac{C_{2}}{C_{3}} = \frac{2 C}{\infty} = 0 \Rightarrow Q_{2} = 0$
Again when C₁ and C₂ are touched
$Q_{2} = 2 \frac{(Q / 3)}{3} \Rightarrow Q_{1} = \frac{(Q / 3)}{3} = \frac{Q}{9}$
Similarly, we can say after N times it becomes $Q_{N} = \frac{Q}{3^{N}}$

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