Concepts Of Physics MCQ Edition [Volume 2]PhysicsCapacitors
A spherical capacitor is formed by two conducting spherical shells having radii a and b . The region between these shells contains a dielectric of dielectric constant K up to a radius c , as depicted in the figure. Determine the capacitance.
Options
- A4 ₀ K a b c a (b - c) + K b(c - a)
- B4 ₀ K a b c K a (b - c) + b(c - a)
- C4 ₀ K a b c K b (c - a) + a(b - c)
- D4 ₀ a b c K a (b - c) + b(c - a)
Correct answer
B. 4 ₀ K a b c K a (b - c) + b(c - a)
Step-by-step solution
The given spherical capacitor can be considered as a combination of two spherical capacitors connected in series: 1. A capacitor C₁ with a dielectric of constant K between the radii a and c . C₁ = 4 ₀ K a c c - a 2. A capacitor C₂ with air between the radii c and b . C₂ = 4 ₀ b c b - c Since the electric field lines pass radially through both regions, the two capacitors are in series. The equivalent capacitance C is given by: 1 C = 1 C₁ + 1 C₂ 1 C = c - a 4 ₀ K a c + b - c 4 ₀ b c 1 C = 1 4 ₀ c ( c - a K a + b - c