Spherical Capacitor Calculator
Enter the inner and outer radii of a spherical capacitor and choose a dielectric material to calculate its capacitance, stored energy, electric field and voltage. Supports reverse-solve modes to find either radius from a target capacitance, and provides a step-by-step derivation with real numbers.
Formula
Worked example
A spherical capacitor with inner radius 5 cm, outer radius 10 cm, filled with air: C = 4*pi*8.854e-12*0.05*0.10/(0.10-0.05) = 11.13 pF. At 100 V, stored energy = 0.5*11.13e-12*100^2 = 55.6 nJ, and the maximum E-field at the inner surface is E(a) = 100*0.10/((0.05)*(0.10-0.05)) = 4000 V/m.
What is a spherical capacitor?
A spherical capacitor consists of two concentric conducting spheres. The inner sphere has radius a, the outer shell has radius b, and the gap between them is filled with a dielectric material of relative permittivity epsilon_r. When a voltage V is applied, equal and opposite charges accumulate on the two surfaces, and the electric field exists only in the gap. Because of the high symmetry of the geometry, the capacitance and field distribution can be solved exactly using Gauss's law, making the spherical capacitor a classical textbook example in electrostatics.
Capacitance formula and derivation
Applying Gauss's law with a spherical Gaussian surface of radius r (a < r < b) gives an electric field E(r) = Q / (4*pi*epsilon_0*epsilon_r*r^2) directed radially outward. Integrating E from a to b gives the potential difference V = Q*(b-a)/(4*pi*epsilon_0*epsilon_r*a*b). Dividing Q by V gives the capacitance: C = 4*pi*epsilon_0*epsilon_r*a*b/(b-a). When the gap is small (b-a much smaller than a), this approaches the parallel-plate formula C = epsilon_0*epsilon_r*A/d with A = 4*pi*a^2 and d = b-a. The relative permittivity epsilon_r multiplies the capacitance linearly - using a material with epsilon_r = 5 gives five times the capacitance of the same geometry in vacuum.
Stored energy and electric field
The energy stored in the electric field of a charged capacitor is U = (1/2)*C*V^2, which is the same formula used for any capacitor geometry. For a spherical capacitor, the field is strongest at the inner sphere surface (r = a) and falls off as 1/r^2 toward the outer shell. The surface field at r = a is E(a) = V*b/((b-a)*a), and at r = b it is E(b) = V*a/((b-a)*b). High-voltage applications must ensure E(a) stays below the dielectric breakdown strength of the fill material. Choosing a dielectric with a high breakdown strength, such as mica or PTFE, allows higher operating voltages without breakdown.
Reverse-solve modes and design use
This calculator supports reverse-solve modes: given a target capacitance and one radius, the other radius is computed directly from the rearranged formula. For the outer radius: b = a/(1 - 4*pi*epsilon*a/C). For the inner radius: a = b/(1 + 4*pi*epsilon*b/C). These modes are useful when designing a capacitor to hit a specific capacitance value within a fixed housing or given only the outer dimension. The dielectric selector lets you explore how filling with mica (er = 5) or alumina (er = 7) changes both the capacitance and the required geometry.
Common dielectric materials and relative permittivity
| Material | Relative permittivity (er) | Dielectric strength (MV/m) |
|---|---|---|
| Vacuum / Air | 1.0 | 3.0 (air) |
| Polytetrafluoroethylene (PTFE) | 2.1 | 19-60 |
| Polyethylene | 2.25 | 18-28 |
| Polystyrene | 2.5 | 20-30 |
| PVC | 3.0 | 20-40 |
| Nylon | 3.7 | 14 |
| Glass (borosilicate) | 4.5 | 14-15 |
| Mica | 5.0 | 100-300 |
| Beryllium oxide | 6.0 | 9-13 |
| Alumina (Al2O3) | 7.0 | 13-17 |
| Silicon | 10 | 30 |
| Ethanol | 25 | ~1.3 |
| Water at 20 C | 80 | ~0.07 |
Typical relative permittivity values at room temperature and standard frequency. Actual values depend on temperature, purity, and frequency.
Frequently asked questions
What is the formula for a spherical capacitor?
The capacitance is C = 4*pi*epsilon_0*epsilon_r*a*b/(b-a), where a is the inner radius, b is the outer radius, epsilon_0 is the permittivity of free space (8.854e-12 F/m), and epsilon_r is the relative permittivity of the dielectric filling the gap. This formula is derived by integrating the electric field (found via Gauss's law) across the gap to find the voltage, then dividing the charge by the voltage.
How does the dielectric material affect capacitance?
The relative permittivity epsilon_r of the fill material multiplies the vacuum capacitance directly. An air-filled capacitor (epsilon_r = 1) has the lowest capacitance for a given geometry. Filling with glass (epsilon_r ~4.5) multiplies it by 4.5, and water (epsilon_r ~80) by 80. However, liquid dielectrics are rarely used in practice due to breakdown and ionic conductivity issues. PTFE, mica, and ceramics are common choices for high-frequency or high-voltage designs.
Where is the electric field strongest inside a spherical capacitor?
The electric field E(r) = V*a*b/((b-a)*r^2) is strongest at r = a, the inner sphere surface, and decreases as 1/r^2 toward the outer shell. This concentration of field at the inner surface is why dielectric breakdown typically initiates there. To reduce peak field, designers increase the inner radius or use a higher-breakdown-strength dielectric.
What happens as the gap becomes very thin?
As b approaches a (thin gap), the spherical capacitor formula approaches the parallel-plate result: C = epsilon_0*epsilon_r*A/d with A = 4*pi*a^2 (inner sphere surface area) and d = b-a (gap). This is a useful limit check: for small gaps compared to the radius, a spherical capacitor behaves like a curved parallel-plate capacitor.
How is stored energy calculated?
Stored energy is U = (1/2)*C*V^2, exactly as for parallel-plate capacitors. Alternatively, U = Q^2/(2C) = (1/2)*Q*V. For the default example (5 cm inner, 10 cm outer, air, 100 V): C = 11.13 pF, U = 0.5*11.13e-12*10000 = 55.65 nJ. Energy scales with the square of voltage, so doubling the voltage quadruples the stored energy.
Can a spherical capacitor have only one conductor?
Yes - an isolated conducting sphere can be treated as a spherical capacitor where the outer shell is at infinity (b approaches infinity). In that limit, the formula simplifies to C = 4*pi*epsilon_0*a, which is the capacitance of an isolated sphere. For a sphere of radius 0.1 m in air, C = 4*pi*8.854e-12*0.1 = 11.13 pF - the same as the default example with b = 2a = 0.1 m and a = 0.05 m.