Problem Statement:
Eight charged water drops, each with a radius of 10mm and charge 10 pico coulomb, merge into a single drop. What is the potential of this single drop?

Solution:

Step 1: Calculate the total charge of the single drop
Given that each individual water drop has a charge of 10 pico coulomb, and there are 8 drops, we can calculate the total charge as follows:

Total charge = Charge per drop × Number of drops
= 10 pico C × 8
= 80 pico C

Step 2: Calculate the radius of the single drop
When the eight drops merge into a single drop, their volumes combine. Since the drops have the same radius, the volume of the single drop can be calculated as:

Volume of single drop = Volume of one drop × Number of drops
= (4/3)πr³ × 8
= (4/3)π(10mm)³ × 8
= (4/3)π(1000mm³) × 8
= (4/3)π(8000mm³)
= (32000/3)π mm³

Now, we can find the radius of the single drop using the formula for the volume of a sphere:

(4/3)πr³ = (32000/3)π
r³ = 32000/3
r = ∛(32000/3)
r ≈ 14.85 mm

Step 3: Calculate the potential of the single drop
The potential of a charged sphere can be calculated using the formula:

Potential = (1/4πε₀) × (Q/R)

Where:
Q = Total charge of the single drop
R = Radius of the single drop
ε₀ = Permittivity of free space (8.85 × 10⁻¹² C²/Nm²)

Plugging in the values:

Potential = (1/4πε₀) × (Q/R)
= (1/4π(8.85 × 10⁻¹² C²/Nm²)) × (80 pico C / 14.85 mm)
= (1/4π(8.85 × 10⁻¹² C²/Nm²)) × (80 × 10⁻¹² C / 0.01485 m)
= (1/4π(8.85 × 10⁻¹² C²/Nm²)) × (8 × 10⁻¹¹ C / 0.01485 m)
= (1/4π(8.85 × 10⁻¹² C²/Nm²)) × (0.571 x 10⁻¹¹ C / m)
= (1/4π(8.85 × 10⁻¹² C²/Nm²)) × (0.571 x 10⁻¹¹ V)

Calculating this expression will give the potential of the single drop in volts.