Spherical Capacitor Calculator

About the Spherical Capacitor Calculator

A spherical capacitor consists of two concentric conducting spheres separated by a dielectric material. Its capacitance depends on the inner radius a, outer radius b, and dielectric constant κ: C = 4πε₀κ(ab)/(b−a). This geometry appears in Van de Graaff generators, high-voltage components, and idealized electrostatics models.

The electric field between the spheres can be derived exactly from Gauss's law: E = Q/(4πε₀κr²). Because the field is strongest at the inner surface and falls as 1/r², the inner conductor is usually the first place to check for dielectric breakdown.

This calculator computes capacitance, stored charge, energy, maximum electric field, and surface charge density, then shows the geometry in an SVG cross-section so you can see how radius and dielectric choice change the result.

When This Page Helps

Spherical capacitors are a compact way to study how geometry changes capacitance and field strength. The formulas are simple enough to write down, but the unit conversions and derived values are easy to slip on if you do them by hand.

This calculator keeps the electrical result, the field check, and the dielectric comparison together so you can inspect the full setup before using it in a design or classroom example.

How to Use the Inputs

1. Select the radius unit (mm, cm, or m) that matches your application.
2. Enter the inner radius (a) of the inner sphere.
3. Enter the outer radius (b) of the outer sphere (must be > a).
4. Select the dielectric material filling the gap between spheres.
5. Enter the applied voltage across the capacitor.
6. Use presets for common configurations like Van de Graaff or lab spheres.
7. Check the dielectric comparison table to see how material choice affects capacitance and energy.

Example Calculation

Tips & Best Practices

• As the gap (b−a) gets very small relative to a, the spherical capacitor approaches the behavior of a parallel plate capacitor with area 4πa².
• The isolated sphere (b→∞) is useful for modeling charged objects far from other conductors — like a metal ball on a stand.
• For water as a dielectric (κ ≈ 80), capacitance increases dramatically — but real water is too conductive for most DC applications.
• Dielectric breakdown occurs when E_max exceeds the material's breakdown strength. Check this before building.
• Mica (κ ≈ 6, breakdown ~100 MV/m) is an excellent dielectric for high-voltage spherical capacitors.
• The Earth and ionosphere form a spherical capacitor with C ≈ 1 farad, charged to ~300,000 V by thunderstorms.

Derivation from Gauss's Law

Applying Gauss's law to a spherical surface of radius r between the conductors: ∮E·dA = Q/ε₀κ, giving E = Q/(4πε₀κr²). Integrating E from a to b gives the potential difference: V = Q/(4πε₀κ) × (1/a − 1/b). Since C = Q/V, we get C = 4πε₀κ(ab)/(b−a). This elegant derivation shows why the spherical capacitor is a classic problem in electrostatics courses.

Real-World Spherical Capacitors

While perfect concentric spheres are rare in commercial electronics, the spherical capacitor model appears in several practical contexts. Van de Graaff generators use a large conducting sphere to accumulate charge, with the dome acting as one electrode and the ground (or a grounded shell) as the other. High-voltage bushings and feedthroughs sometimes use spherical geometries to minimize electric field concentration.

Relationship to Other Capacitor Geometries

The three analytically solvable capacitor geometries — parallel plate, cylindrical, and spherical — form a hierarchy of increasing mathematical complexity. The parallel plate (C = ε₀A/d) has a uniform field, the cylindrical (C = 2πε₀L/ln(b/a)) has a 1/r field, and the spherical has a 1/r² field. Understanding all three provides deep insight into electrostatic energy storage.

Frequently Asked Questions

• Why is the inner sphere more prone to breakdown?

  The electric field is E = Q/(4πε₀κr²), so it is inversely proportional to r². The smallest radius (inner surface) has the highest field concentration, making it the first location to exceed the dielectric breakdown strength.

• What happens if I set b to infinity?

  The formula simplifies to C = 4πε₀κa — the capacitance of an isolated sphere. This is a useful concept in electrostatics: even a single conductor has capacitance with respect to infinity.

• How does a spherical capacitor compare to a parallel plate capacitor?

  For the same gap and surface area, a spherical capacitor has a non-uniform electric field (stronger near the inner sphere), while a parallel plate has a uniform field. When the gap is much smaller than the radius, the two converge.

• Can I use this for a hemisphere or partial sphere?

  This calculator assumes full concentric spheres. For a hemisphere, the capacitance is approximately half. For partial spheres or non-concentric geometries, numerical methods (FEM) are typically needed.

• What is the energy density inside the capacitor?

  The energy density is u = ½ε₀κE² and varies with position (r). It is highest at the inner surface and decreases as 1/r⁴. The total energy is the integral over the volume: U = ½CV².

• Why does water have such a high dielectric constant?

  Water molecules are strongly polar (permanent dipole moment). In an electric field, they orient to oppose the field, effectively reducing it and increasing capacitance. This high κ = 80 is why water is such a good solvent for ionic compounds.