Twenty seven drops of same size are charged at 220 V each. They combine to form a bigger drop. Calculate the p

Question

Twenty seven drops of same size are charged at 220 V each. They combine to form a bigger drop. Calculate the potential of the bigger drop:

Accepted Answer

1. **Conservation of Volume:** Let $r$ be the radius of a small drop and $R$ be the radius of the big drop. The total volume remains constant during the combination.
   $\frac{4}{3}\pi R^3 = 27 	imes \frac{4}{3}\pi r^3 \implies R = (27)^{1/3} r = 3r$.
2. **Conservation of Charge:** Let $q$ be the charge on a small drop. The total charge $Q$ on the big drop is the sum of the charges on the 27 small drops.
   $Q = 27q$.
3. **Potential Relationship:** The electric potential $V$ of a conducting sphere is given by $V = \frac{kQ}{R}$ (analogous to the gravitational potential $V \propto \frac{M}{R}$ discussed in Class 11 Physics, Chapter 8).
   - Potential of small drop: $V_{small} = \frac{kq}{r} = 220 	ext{ V}$.
   - Potential of big drop: $V_{big} = \frac{kQ}{R} = \frac{k(27q)}{3r} = 9 \left( \frac{kq}{r} 
ight) = 9 V_{small}$.
4. **Calculation:**
   $V_{big} = 9 	imes 220 	ext{ V} = 1980 	ext{ V}$.

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