# Capacitance of an Isolated Sphere(OCR A Level Physics)

Author

Katie M

Expertise

Physics

## Capacitance of an Isolated Sphere

• The capacitance, C, of a charged sphere, is defined as the charge per unit potential at the surface of the sphere

• Where:
• C = capacitance (F)
• Q = charge (C)
• V = potential difference (V)

• The charge on the surface of a spherical conductor can be considered as a point charge at its centre
• The potential V of an isolated point charge is given by:

• Where:
• R = radius of sphere (m)
• ε0 = permittivity of free space
• The charge, Q, is not the charge of the capacitor itself, it is the charge stored on the surface of the spherical conductor
• Combining these equations gives an expression for the capacitance of an isolated sphere:

C = 4πε0R

#### Worked example

Lightning can be simulated in a laboratory using an isolated metal sphere to investigate electrical discharge.

A sphere of radius 75 cm is charged to a potential of 1.5 MV.

Following the electrical discharge, the sphere loses 95% of its energy.

Calculate:

a)
The capacitance of the sphere.
b)
The potential of the sphere after discharging.

Part (a)

Step 1: List the known quantities

• Radius of sphere, R = 75 cm = 75 × 10−2 m
• Permittivity of free space, ε0 = 8.85 × 10−12 F m−1

Step 2: Write out the equation for the capacitance of a charged sphere

C = 4πε0R

Step 3: Calculate the capacitance

C = 4π × (8.85 × 10−12) × (75 × 10−2)

C = 8.34 × 10−11 F

Part (b)

Step 1: List the known quantities

• Original potential, V1 = 1.5 MV = 1.5 × 106 V
• Final potential = V2
• Original energy = E1
• Final energy, E2 = 0.05 E1

Step 2: Write out the equation for the energy stored by a capacitor

Step 3: Write out equations for energy before and after discharge

Step 4: Equate the two expressions and simplify

• Since E2 = 0.05 E1

Step 5: Calculate the final potential, V2

V2 × (1.5 × 106) = 3.35 × 105 V

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