Electrostatics and Magnetostatics
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 :
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 :
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 :
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
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 :