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

• A. 1980 V
• B. 660 V
• C. 1320 V
• D. 1520 V

Answer 1

The Correct Option is A

The problem involves finding the potential of a larger spherical drop formed by the coalescence of 27 smaller drops, each charged to 220 V. We will solve this using the concept of charge conservation and the formula for potential due to a spherical conductor.

1. Determine the charge of the smaller drops: \(V = \frac{k \cdot q}{r}\) where V is the potential, k is the Coulomb's constant, q is the charge, and r is the radius of the drop. Since the potential V and the radius r are the same for all the drops, the charge q of each drop is proportional to its potential.
2. Use charge conservation: All 27 smaller drops combine to form a bigger drop, so the total charge \(Q\) of the bigger drop is the sum of the charges of the 27 individual drops. \[ Q = 27q \]
3. Determine the volume of the bigger drop: The volume of a sphere is given by \( V = \frac{4}{3}\pi r^3 \). Since volume is conserved, the volume of the larger drop equals the sum of the volumes of the smaller drops, \[ \frac{4}{3}\pi R^3 = 27 \times \frac{4}{3}\pi r^3 \] On simplification, we find \[ R = 3r \]
4. Calculate the potential of the bigger drop: The potential of a spherical conductor is given by: \[ V' = \frac{k \cdot Q}{R} \] Substituting the values of \( Q \) and \( R \), \[ V' = \frac{k \cdot 27q}{3r} = 9 \cdot \left(\frac{k \cdot q}{r}\right) = 9 \times 220 = 1980 \, \text{V} \]

Thus, the correct answer is 1980 V. This is consistent with the principle that the potential scales with the charge and inversely with the radius for a spherical conductor.