Capacitance of an Isolated Sphere (OCR A Level Physics): Revision Note

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.

Answer:

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

$E=\frac{1}{2}C{V}^2$

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

$E_1=\frac{1}{2}C{V_1}^2$

$E_2=\frac{1}{2}C{V_2}^2$

Step 4: Equate the two expressions and simplify

• Since E₂ = 0.05 E₁

$\frac{1}{2}C{V_2}^2=0.05\left(\frac{1}{2}C{V_1}^2\right)$

${V_2}^2=0.05\left({V_1}^2\right)$

$V_2=\sqrt{0.05}{V}_1$

Step 5: Calculate the final potential, V₂

V₂ = $\sqrt{0.05}$ × (1.5 × 10⁶) = 3.35 × 10⁵ V