Table of Contents
What Is a Spherical Capacitor?
A spherical capacitor consists of two concentric conducting spheres separated by a dielectric material. The inner sphere of radius a carries charge +Q, while the outer sphere of radius b carries charge -Q. The electric field exists only in the region between the spheres, making this geometry ideal for studying electrostatics and Gauss's law.
Spherical capacitors appear in Van de Graaff generators, certain sensor designs, and theoretical physics problems. Earth itself acts as a spherical capacitor with the ionosphere as the outer conductor, with a capacitance of approximately 710 microfarads.
Capacitance Formula
Where ε0 = 8.854 × 10^-12 F/m is the permittivity of free space, κ is the relative permittivity (dielectric constant), a is the inner radius, and b is the outer radius. For an isolated sphere (b → ∞), the capacitance simplifies to C = 4πε0κa.
Dielectric Constants
| Material | Relative Permittivity (κ) |
|---|---|
| Vacuum | 1.0 (exact) |
| Air | 1.0006 |
| Teflon (PTFE) | 2.1 |
| Polyethylene | 2.25 |
| Paper | 3.0-4.0 |
| Glass | 4-10 |
| Mica | 5-7 |
| Water (20°C) | 80.1 |
| Barium titanate | 1,200-10,000 |
Special Cases
- Isolated sphere (b → ∞): C = 4πε0a. Earth's self-capacitance is about 710 μF.
- Thin shell (b ≈ a): Approaches a parallel-plate capacitor with C = ε0κA/d where A = 4πa² and d = b - a.
- Concentric cylinders: Different geometry uses C = 2πε0κL / ln(b/a).
Frequently Asked Questions
How does increasing the gap affect capacitance?
Increasing the gap (b - a) decreases capacitance because the capacitance formula has (b - a) in the denominator. For a fixed inner radius, a larger outer radius means more separation, weaker electric field, and less charge stored per volt.
What is the electric field inside a spherical capacitor?
By Gauss's law, E = Q / (4πε0κr²) for a < r < b. The field points radially outward (for positive charge on the inner sphere) and decreases with the square of the distance. Inside the inner sphere (r < a) and outside the outer sphere (r > b), the field is zero.
Can a single sphere be a capacitor?
Yes. An isolated conducting sphere has self-capacitance C = 4πε0a. This represents the charge it can hold per volt relative to infinity. A 1-meter radius sphere in vacuum has a capacitance of about 111 picofarads.