82. Spherical capacitance. Units of electrical capacitance

Let's calculate the capacitance of the sphere of radius R. The potential of the sphere in a uniform dielectric is equal (see section 77)

\( V \,= \,\frac{Q}{4\pi\,\varepsilon{_0}\,\varepsilon\,R} \)

(potential at infinity is considered zero).

On the other hand, according to the definition of capacitance

\( V \,= \,\frac{Q}{C} \)

By comparing these equations, we obtain

\( C \,= \,4\pi\,\varepsilon{_0}\,\varepsilon\,R \) (8-27)

The capacitance of the sphere is proportional to its radius. It does not depend on the sphere material. All that matters is that it is conductive.

The potential of the sphere in vacuum is determined by equation

\( V \,= \,\frac{Q}{R} \)

Accordingly, in vacuum, the sphere capacitance is numerically equal to the sphere radius

\( C \,= \,R \) (8-28)

The unit of capacitance in the International System of Units is the capacitance of such a conductor, in which there appears a potential difference of \(1\,V\) volt when it is charged by a quantity of electricity of \(1\,q\) coulomb. This unit is called a farad.

It is not difficult to see that \(1\,f \,= \,\frac{1\,q}{1\,V}\).

The capacitance in one farad is very large. That is why in practice the fractions of this unit are often used: microfarad \((mkf) \,10^{-6} \,farades\) and picofarada \((pkf) \,10^{-12} \,farades\).

The Earth's capacitance as a sphere is equal to \(709 \,mcf\). A sphere with a capacitance of \(1\,f\) would have a radius 13 times greater than the radius of the Sun.

Note that according to \((8-27)\) the electric constant can be measured in faradas per meter \((f/m)\).