# Formula: **Plate Capacitor** Capacitance Plate area Plate distance Relative permittivity

## Level 3

Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

`\[ C ~=~ \varepsilon_0 \, \varepsilon_{\text r} \, \frac{A}{d} \]`

`\[ C ~=~ \varepsilon_0 \, \varepsilon_{\text r} \, \frac{A}{d} \]`

`\[ A ~=~ \frac{C \, d}{ \varepsilon_0 \, \varepsilon_{\text r} } \]`

`\[ d ~=~ \varepsilon_0 \, \varepsilon_{\text r} \, \frac{A}{C} \]`

`\[ \varepsilon_{\text r} ~=~ \frac{C \, d}{ \varepsilon_0 \, A } \]`

`\[ \varepsilon_0 ~=~ \frac{C \, d}{ \varepsilon_{\text r} \, A } \]`

**Rearrange formula**

## Capacitance

`\( C \)`Unit

`\( \text{F} \)`

Capacitance is a measure of how much charge the plate capacitor can "store".

As can be seen from the formula, the capacitance depends only on the geometry of the plate capacitor, i.e. only on the plate area \(A\) and the distance \(d\) between the plates.

## Plate area

`\( A \)`Unit

`\( \text{m}^2 \)`

Plate area is the area of one side of the capacitor plate. For a rectangular plate, \(A\) is the area of a rectangle: \(A = a \, b \). For a circular electrode, \(A\) is the area of a circle: \(A = \pi \, r^2 \).

The larger the plate area, the greater the capacitance of the plate capacitor.

## Plate distance

`\( d \)`Unit

`\( \text{m} \)`

Distance between the two capacitor plates. The closer the plates are to each other, the greater the capacitance of the plate capacitor.

## Relative permittivity

`\( \varepsilon_{\text r} \)`Unit

`\( - \)`

Relative permittivity is a dimensionless number that describes the dielectric (e.g. air, water, glass) between the two capacitor plates.This number indicates how well the dielectric transmits the electric field between the plates.

In a vacuum the relative permittivity has the value \( \varepsilon_{\text r} = 1 \). Water at room temperature: \( \varepsilon_{\text r} \approx 1.77 \). Glass: \( \varepsilon_{\text r} \approx 7 \). Consequently, a dielectric which allows the E-field to pass through less well (relative permittivity is high) increases the capacitance of the plate capacitor.

## Vacuum permittivity

`\( \varepsilon_0 \)`Unit

`\( \frac{ \text{As} }{ \text{Vm} } \)`

Vacuum permittivity is a physical constant that always occurs with electric phenomena. The constant has the following value: \( \varepsilon_0 = 8,854 \cdot 10^{-12} \, \frac{ \text{As} }{ \text{Vm} } \).