Electromagnetism - Maxwell's Equations & Circuit Theory

The electric potential energy per unit charge at a point in space: $V=\frac{U}{q}$. The relationship with electric field is $\vec{E}=-\nabla V$.

Describes the electrostatic force between two stationary, point charges: $F=k_e\frac{|q_1{q}_2|}{r^2}$, where $k_e=\frac{1}{4\pi {ϵ}_0}\approx 8.99\times 10^9\text{ N}\cdot {\text{m}}^2/{\text{C}}^2$.

The electric force per unit positive charge exerted on a test charge at a given point: $\vec{E}=\frac{\vec{F}}{q_0}$. Measured in volts per meter (V/m) or N/C.

Maxwell's 1st equation: $\oint \vec{E}\cdot d\vec{A}=\frac{Q_{enc}}{{ϵ}_0}$. States that the net electric flux through any closed surface is proportional to the total enclosed charge $Q_{enc}$.

Maxwell's 2nd equation: $\oint \vec{B}\cdot d\vec{A}=0$. States that the net magnetic flux through any closed surface is always zero, implying magnetic monopoles do not exist.

Maxwell's 3rd equation: $\mathcal{E}=-\frac{d{\Phi }_B}{dt}$ or $\oint \vec{E}\cdot d\vec{l}=-\frac{\partial }{\partial t}\iint \vec{B}\cdot d\vec{A}$. A changing magnetic flux induces an electromotive force (EMF) and electric field.

States that the direction of an induced current always produces a magnetic field that opposes the change in magnetic flux that created it (represented by the negative sign in Faraday's Law).

Relates the circulating magnetic field around a closed loop to the electric current passing through the loop: $\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{enc}$.