Electric Field and Gauss's Law

Electric Fields and Gauss's Law

Instead of tracking forces charge by charge, imagine every charge filling space with an invisible electric field: a ready-made instruction at every point saying which way a test charge would be pushed.

The electric field $\vec{E}$ is the force per unit charge at a point in space:

$$\vec{E} = \frac{\vec{F}}{q_0} = k_e\frac{Q}{r^2}\hat{r}$$

Electric field lines start on positive charges, end on negative charges, never cross, and have a density proportional to field strength.

Gauss's law is the field's conservation statement, relating the electric flux through any closed surface to the enclosed charge:

$$\oint_S \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$

Gauss's law is one of Maxwell's four equations and is equivalent to Coulomb's law for electrostatics.

Gaussian Surfaces and Differential Form

Choosing a Gaussian surface: Exploit symmetry so that $\vec{E}$ is constant on the surface and easy to pull out of the integral.

The differential form of Gauss's law links the field's divergence to the local charge density $\rho$:

$$\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0}$$

Common pitfall: Zero flux through a closed surface does not mean zero field on it. An external charge sends lines in and out; they cancel in the count, yet the local field is non-zero.