Question

' n ' small spherical drops of same size which are charged to ' $V$ ' volt each coalesce to form a single big drop. The potential of the big drop is

• A. $\frac{V}{n}$
• B. $n \cdot V$
• C. $n^{1/3} \cdot V$
• D. $\text{n}^{2/3} \cdot \text{V}$

Hint:

For $n$ drops coalescing: Radius $\propto n^{1/3}$, Charge $\propto n$, Potential $\propto n^{2/3}$, and Capacity $\propto n^{1/3}$.

Answer 1

The Correct Option is D

Step 1: Concept
Potential of a single small drop is $v = \frac{q}{4\pi\varepsilon_0 r}$. For the big drop, charge $Q = nq$ and volume is conserved.

Step 2: Meaning
Conservation of volume: $\frac{4}{3}\pi R^3 = n \frac{4}{3}\pi r^3 \implies R = n^{1/3}r$.

Step 3: Analysis
Potential of big drop $V_{big} = \frac{Q}{4\pi\varepsilon_0 R} = \frac{nq}{4\pi\varepsilon_0 (n^{1/3}r)}$. $V_{big} = \frac{n}{n^{1/3}} \cdot \frac{q}{4\pi\varepsilon_0 r} = n^{2/3}V$.

Step 4: Conclusion
The potential of the big drop is $n^{2/3}V$. Final Answer: (D)