Spherical Capacitor Calculator

Created by Dominik Czernia, PhD candidate
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Sep 06, 2022

This spherical capacitor calculator will help you to find the optimal parameters for designing a spherical capacitor with a specific capacitance.

Unlike the most common parallel-plate capacitor, spherical capacitors consist of two concentric spherical conducting shells separated by a dielectric.

Read on to learn about the capacitors, the spherical capacitor equation, and about two combinations of spherical capacitors.

What is a capacitor?

A capacitor is one of the essential elements of the electrical circuit which can store and release electric charge.

Capacitors are widely used in many electronic devices to perform a variety of tasks, such as smoothing, filtering, or bypassing an electrical signal. The construction of the capacitor is straightforward – they mainly consist of two separate plates.

To determine the amount of electric charge that a capacitor can store, we use a quantity called capacitance. The more electric charge capacitor can hold, the higher its capacitance. To further improve capacitance, we can fill the space between plates with some nonconducting dielectric medium.

Spherical capacitor with dielectric equation

You can calculate the capacitance of a spherical capacitor using the following formula:

C=4πε0εk1/a1/b\small C = \frac{4 \pi \varepsilon_0 \varepsilon_k}{1/a - 1/b}

where:

  • CC – Capacitance measured in farads (symbol: F);
  • ε0\varepsilon_0 – Vacuum permittivity – a constant value of 8.85×1012 F/m\small 8.85 \times 10^{−12}\ \text{F/m} (farads per meter);
  • εk\varepsilon_k – Relative permittivity (it is dimensionless);
  • aa – Radius of the inner sphere; and
  • bb – Radius of the outer sphere.

The relative permittivity εk\varepsilon_k is a constant, characteristic for a specific dielectric placed between the capacitor plates. In our spherical capacitor calculator, we assumed for simplicity that there is a vacuum between plates and, therefore, εk=1\varepsilon_k = 1. If you want to change the material located between plates, go to the advanced mode of the spherical capacitor calculator.

Spherical capacitors in parallel or series

Spherical capacitors can be combined in parallel and series, too!

  • Imagine that our capacitor consists of three concentric spheres where spaces between them are filled with different dielectrics. We can treat those spaces like separate capacitors combined in series, and we can calculate the total capacitance in the same way as parallel resistors.

  • Now imagine that our capacitor consists of two concentric spheres, but the space between them is divided into two halves, in which the space between shells is filled with different dielectrics. We can treat both halves as separate capacitors combined in parallel, and the total capacitance can be calculated in the same way as resistors in series.

Dominik Czernia, PhD candidate
Inner sphere radius
in
Outer sphere radius
in
Capacitance
pF
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