Gauss’s law - electric field due to an infinite wire of charge

In this page, we are going to see how to calculate the magnitude of the electric field due to an infinite wire of positive charge at a distance r from the wire using Gauss’s law. The result has to be the same as obtained calculating the electric field due to an infinite wire using Coulomb’s law.

Gauss’s law gives a value to the flux of an electric field passing through a closed surface:

Where the sum in the second member is the total charge enclosed in the surface.

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In order to apply Gauss’s law, we first need to draw the electric field lines due to a continuous distribution of charge, in this case an infinite wire. We also need to choose the Gaussian surface through which we will calculate the flux of the electric field.

The wire of charge and the lines of the electric field due to it are represented in the next figure. These field lines emerge radially from the wire because the positive charges are sources of field lines.

The Gaussian surface (a cylinder of radius r) through which we are going to calculate the flux of the electric field is represented in red in the figure. The vectors dS for each one of the flat bases of the cylinder (with a surface area S) and for the lateral curved side are represented in red in the same figure.

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The net flux through the surface is:

The flux through the flat bases of the cylinder is zero because, as you can see in the figure, there is no field lines passing through them. Therefore, the only contribution to the flow is from the lateral curved side of the cylinder. Vectors E and dS are parallel for this side, so we have:

Where l is the height of the cylinder.

The linear charge density is by definition:

Therefore we have:

This is the same result as obtained calculating the electric field due to an infinite wire using Coulomb’s law.

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