Kirchhoff's Laws

First Law

$$\Sigma I=0$$

In an intersection (A node where 3 or more wires intersect).

Note

$I$ here is signed. Positive for incomming current, and negative for outgoing, or the inverse: as long as its consistant.

Second Law

$$\begin{array}{rl}& \Sigma V=0\\ & {\oint }_{\Gamma }E\cdot ds=0\end{array}$$

Around a closed loop, since electrostatic force is conservative.

Faraday-Henry-Lenz Law

When there is a non-uniform, the integral form does not apply. We therefore must modify it to:

$${\oint }_{\Gamma }\vec{E}(\vec{r})\cdot d\vec{s}=-\frac{d{\Phi }_B}{dt}=\frac{W}{q}=\text{emf}$$

Meaning that Magnetism and Electric fields are linked.

EMF is the work done per unit charge by an electric field generated by a time-dependant magnetic flux.