Capacitance of a dielectric capacitor with two sphere electrodes instead of parallel plates

Question

EDIT: title changed to be more descriptive.

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I would like to calculate the capacitance of a material (in plate form with thickness d and a known permittivity $\varepsilon_r$ ) between two sphere electrodes of the same radius.

I could calculate the capacitance using the plate model of a capacitor: $$ \frac {\varepsilon_0 \varepsilon_r A}{d}$$ but how do I determine the effective area A between the sphere electrodes?

Answer 1

For two identical sphere electrodes of radius $a$:

$$C=\frac{2 \pi \epsilon_{0} \epsilon_{r} a}{1-\left(\frac{a}{d-a}\right)}$$

where $ε_{r}$ is the dielectric constant or relative permittivity of your dielectric material, $\epsilon_{0}$ the permittivity of vacuum space and $d$ the axial distance between the centers of the two spheres. The derivation is complicated and demonstrated in this video here:

https://www.youtube.com/watch?v=k8Sp7r3rQBc

Assuming parallel plates your initial electric field $E_{0}$ on air would be:

$$E_{0}=\frac{V}{d}$$

where $V$ the applied voltage on the plates.

Therefore the final $E$ field value in your dielectric center will be:

$$E=\frac{E_{0}}{k}$$

where kappa is the dielectric constant or relative permittivity of your dielectric material $ε_{r}$.