84. Plate capacitor

Let's calculate the capacitance of the flat capacitor. First of all, we calculate the strength of the electric field inside the charged capacitor. Let the area of the capacitor plates be \(A\), the distance between them \(d\). The capacitor is filled with a dielectric material with dielectric permittivity \(\varepsilon\).

The electrical field in the capacitor is summed up from the fields of positively and negatively charged plates. Since the distance between the plates is much smaller than their size, we can use equation \((8-17)\)

\( E \,= \,\frac{\sigma}{2\,\varepsilon{_0}\,\varepsilon} \,+ \,\frac{\sigma}{2\,\varepsilon{_0}\,\varepsilon} \,= \,\frac{\sigma}{\varepsilon{_0}\,\varepsilon} \)

The surface charge density is

\( \sigma \,= \,\frac{Q}{A} \)

Therefore,

\( E \,= \,\frac{Q}{\varepsilon{_0}\,\varepsilon\,A} \)

The potential difference of a uniform capacitor field is expressed by equation \((8-25)\)

\( \Delta{V} \,= \,E\,d \,= \,\frac{Q\,d}{\varepsilon{_0}\,\varepsilon\,A} \) (8-30)

(Here \(\Delta{V}\) is the absolute value of the potential difference and instead of a small distance of \(\Delta{d}\), the distance \(d\) as final is taken).

By adding \((8-30)\) in \((8-29)\), we obtain

\( C \,= \,\frac{\varepsilon{_0}\,\varepsilon\,A}{d} \)

The capacitance of the flat capacitor is directly proportional to the plate area and the dielectric permittivity and is inversely proportional to the distance between the plates.

As the distance \(d\) can be made very small, the area \(A\) and the dielectric permittivity are large enough, the capacitance of the capacitor is large at small geometric dimensions.

A capacitor the size of a book can have a capacitance larger than the earth! As a result, capacitors store huge amounts of charge at relatively small potential differences between the plates.

The accumulation of a large charge through a large capacitance rather than voltage is very important. The voltage cannot be increased infinitely, as a capacitor will break down. Its plates will short-circuit and the capacitor will fail.