Twenty seven spherical drops of radius 3 mm and carrying 10⁻¹² C of...

Question

Twenty seven spherical drops of radius 3 mm and carrying 10⁻¹² C of charge each are combined to form a single drop. Find the capacitance and potential of the bigger drop.

Accepted Answer

Given:
• Number of drops, n = 27
• Radius of each small drop, r = 3 mm = 3 × 10⁻³ m
• Charge on each drop, q = 10⁻¹² C

Step 1 — Radius of Bigger Drop (Volume Conservation):
n × (4/3)πr³ = (4/3)πR³
R³ = n × r³ = 27 × (3×10⁻³)³
R³ = 27r³
R = 3r = 3 × 3 × 10⁻³
R = 9 × 10⁻³ m = 9 mm

Step 2 — Total Charge:
Q = n × q = 27 × 10⁻¹² = 27 × 10⁻¹² C

Step 3 — Capacitance of Bigger Drop:
C = 4πε₀R = R / (9 × 10⁹)
C = (9 × 10⁻³) / (9 × 10⁹)
C = 10⁻¹² F = 1 pF

Step 4 — Potential of Bigger Drop:
V = Q / C = (27 × 10⁻¹²) / (10⁻¹²)
V = 27 V

∴ Capacitance = 1 pF and Potential = 27 V

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