Electric field in a cylindrical capacitor

A capacitor is a device used in electric and electronic circuits to store electrical energy as an electric potential difference (or in an electric field). It consists of two electrical conductors (called plates), typically plates, cylinder or sheets, separated by an insulating layer (a void or a dielectric material). A dielectric material is a material that does not allow current to flow and can therefore be used as insulator.

The first capacitor was build in 1745-1746 and consisted of a glass jar covered by metal foil on the inside and outside. It is known as the Leyden jar (or Leiden jar).

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In this page we are going to calculate the electric field in a cylindrical capacitor.

A cylindrical capacitor consists of two cylindrical concentric plates of radius R1 and R2 respectively as seen in the next figure. The charge of the internal plate is +q and the charge of the external plate is –q.

The electric field created by each one of the cylinders has a radial direction. The field lines are directed away from the positive plate (in green) and toward the negative plate. We are going to use Gauss’s law to calculate the magnitude of the electric field between the capacitor plates. The electric field inside the cylinder of radius R1 or outside the capacitor is zero.

The capacitor and the Gaussian surface (a cylinder of radius r in red dashed lines) we will use to calculate the flux are represented viewed from the top in the next figure.

The flux through the Gaussian surface is given by:

The flux through the cylinder bases is zero, because no field lines pass through them. The vectors E and dS are parallel for the lateral surface of the cylinder, therefore their dot product is equal to the product of their magnitudes. Furthermore, as the field has a radial symmetry, its magnitude is the same for all the points lying on the Gaussian surface and we can therefore move it outside the integral:

And after substituting the surface of the lateral side of the cylinder we obtain:

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The capacitance C of a capacitor is defined as the ratio between the absolute value of the plates charge and the electric potential difference between them:

The SI unit of capacitance is the farad (F).

First, we are going to calculate the electric potential difference between the capacitor plates:

The vectors we will need to perform are represented in the figure below:

After substituting in the electric potential difference we get:

And after integrating we obtain:

The capacitance of the capacitor is therefore:

During the charge of a capacitor, a positive charge dq is transferred from the negative plate to the positive one. But, in order to do that, it is necessary to provide a certain amount of energy in the form of work, because if it were not the case, the positive charge would be repelled by the negative plate.

The work done to move the charge dq from the negative to the positive plate is given by:

We integrate between an empty charge and the maximal charge q to obtain:

If we express q as a function of the capacitor’s capacitance we have:

The energy used to charge the capacitor stays stored in it.

Therefore, the energy stored in a charged capacitor is:

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