Applications of Gauss' Theorem > Electric Field due to a Point Charge

Gauss' theorem provides a shortcut method to calculate electric fields in situations with high symmetry — such as spherical, cylindrical, or planar charge distributions — avoiding lengthy integration used in Coulomb's law-based calculations.

For a point charge, the natural choice of Gaussian surface is a sphere centered on the charge, since the electric field has the same magnitude at every point on such a surface and is directed radially outward (or inward for negative charge).

Setup: A point charge +q is placed at the origin O. A spherical Gaussian surface of radius r is drawn around it, centered at O. Point P lies on this surface.

Step 1: Apply Gauss' Law

Step 2: Express flux through the sphere
Since E is constant in magnitude and parallel to the area vector at every point on the sphere:

Step 3: Equate both expressions

Step 4: Solve for E

Direction: radially outward for +q, radially inward for −q.

Placing a small test charge q_0​ at point P, the force experienced is:

F = \[q_0E=\frac{1}{4\pi\epsilon_0}\frac{qq_0}{r^2}\]

This is exactly Coulomb's Law — proving that Coulomb's law is a direct consequence of Gauss' theorem, not an independent postulate.

Think of the point charge as a sprinkler spraying water equally in all directions. Imagine an invisible balloon (Gaussian surface) around the sprinkler — the total amount of water passing through the balloon's surface per second is the same no matter how big the balloon is, but the water hitting each square centimeter (the "intensity") decreases as the balloon gets bigger. This mirrors how electric flux stays constant while field intensity E falls off as 1/r².