Gauss' Law

For any closed surface $S$ enclosing a set of charges $Q\_enclosed=\sum \_{i=1}^n(q\_n)$ Gauss’ law generalises the flux (in the given context the total charge towards the outside of the enclosed region) as not dependent on $S$ as long as it encloses the charges.

$$\oint E\cdot da=\frac{1}{ϵ\_0}Q\_\text{enclosed}$$

Differential Form

By applying the Theorem for Divergence and equating the resulting expression with $Q\_\text{enclosed}$ defined in terms of the charge density $\rho$, Gauss’ law in differential form is derived.

$$\nabla \cdot E=\frac{1}{ϵ\_0}\rho$$

Remarks

• If there is more charge outside the spanned area, it will contribute to the field, but not to the flux integral.