Capacitance of an Isolated Sphere (OCR A Level Physics)

• 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

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