Chapter 1

# Capacitors

### Définition : Definition of a capacitor

A set of two conductors, one of which is empty and surrounds the other completely is called a capacitor.

The space which separates the two frames can either be empty or filled with an insulator (“dielectric”).

The facing frames carry opposite charges, noted and .

The capacitance of the capacitor is defined by :

Definition of a capacitor

### Fondamental : Energetic aspect, plane capacitor example

In electricity, we have seen that the energy of a capacitor could be written as such :

is the voltage difference between the terminals of the capacitor.

For a plane capacitor, the electric field inside the conductive plates is even and is equal to :

Where is the charge of the positive plate and its surface.

The tension is obtained by the line integral around a loop of the field between the two conductive plates :

By calling the distance between the two plates.

Thus we find the capacitance of a plane capacitor :

We can also see that the energy of a capacitor can be written as such :

Thus :

By noticing that represents the volume between the two frames, the expression of the density of purely electric energy (the electric energy per unit of volume) is :

The expression of this energy is specified in the lesson about electromagnetic energy review.

### Exemple : Calculation of the capacitance

Spherical capacitor :

The study of symmetry and invariance gives :

Spherical capacitor

Gauss' law applied to a sphere of center O and radius leads to :

To determine the voltage difference between the two frames, let's calculate the line integral of the electric field between the two frames :

Where and  are the radius of the two electrodes ( ).

Hence we can deduce the capacitance of the spherical capacitor :

Cylindrical capacitor :

The method used is identical to the one used for the spherical capacitor

Cylindrical capacitor

The electric field between the two frames is given by Gauss' Law :

Let's compute the line integral of the field between the two frames :

Where and are the radius of the two frames ( ).

Hence the capacitance :

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