Answer
🔋 Capacitance of Spherical Capacitor with Radially Varying Dielectric
🧮 Part (a): Derivation
The permittivity varies with radial distance \( r \) as:
ε(r) = ε₀ (1 + k / (r – R₁))
The electric field from Gauss’s law is:
E(r) = Q / [4πr²ε(r)] = Q(r – R₁) / [4π ε₀ r² (r – R₁ + k)]
🔻 Potential Difference:
Using integration from R₁ to R₂:
V = ∫R₁R₂ E(r) dr = (Q / 4πε₀) ∫0R₂−R₁ [x / (x + R₁)²(x + k)] dx
After change of variables \( x = r – R₁ \), the expression for Capacitance becomes:
C = 4πε₀ / ∫0R₂−R₁ [x / (x + R₁)²(x + k)] dx
🧪 Part (b): Numerical Evaluation
| Parameter | Value |
|---|---|
| R₁ | 1.00 cm = 0.01 m |
| R₂ | 3.00 cm = 0.03 m |
| k | 0.50 cm = 0.005 m |
| ε₀ | 8.854 × 10⁻¹² F/m |
Approximating the integral numerically:
∫ ≈ 44.24
Then the capacitance is:
C ≈ (4π × 8.854 × 10⁻¹²) / 44.24 ≈ 2.52 × 10⁻¹² F = 2.52 pF
✅ Final Answer
Capacitance Expression:
C = 4πε₀ / ∫0R₂−R₁ [x / (x + R₁)²(x + k)] dx
Numerical Result:
C ≈ 2.52 picofarads (pF)