Gauss's Law for a Single Charge

Gauss's Law applies to any charge contribution, but let us apply it now to the simplest case, that of a single point charge q.

Construct a spherical Gaussian surface of radius r around the charge q. Take a small area on the Gaussian surface. The area vector points radially outward, as does the electric field vector at this point. Therefore, the electric flux through this small area is

From the spherical symmetry, all of such small area elements contribute equally to the total.

According to Gauss's Law,

because = q. Solving for the electric field gives

which is just Coulomb's Law!

When you're ready, go on to learn about how symmetry plays a role.


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