Definition: Coulomb's Law

Coulomb’s Law says that the electric field from a point charge is given by $$\mathbf E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}\hat{\mathbf r}.$$

Definition: Gauss's Law

Gauss's Law says that $$\Phi_E = \oint \mathbf E \cdot d\mathbf A = \frac{q_{\text{enc}}}{\epsilon_0}.$$

Tip: Electric Field Lines

A visually helpful tool when working with electrostatics is electric field lines. These are lines which follow the electric field vectors tangentially, meaning that electric potential is monotonically decreasing along them (electric potential will be discussed further in the second electrostatics lecture). Field lines can originate from either a positive charge or infinity and must end on either a negative charge or go to infinity.

Theorem: Force on a Layer of Charge

The force on a layer of charge of charge density $\sigma$ is $$F = \frac 12 (E_1 + E_2)\sigma A,$$where $E_1$ and $E_2$ are the charge densities on the two sides of the layer of charge.

Tip: Symmetry Tricks!

• You can utilize non-spherical symmetries!
• Use Newton’s Third Law — find the force on the more convenient of two objects.
• You can deform your flux surface as Gauss’s law on empty space gives 0 flux
• You can “project pressure” onto more convenient surfaces. That is, if you have some integral $\int P_0 d\mathbf A$, along a surface with some boundary curve $C$, then we can use any surface which has boundary curve $C$ for the integral.