Electromagnetism (classical)

Maxwell’s equations

  • Electric field \(\vec{E}\) produced by charge \(q\).
  • Magnetic field \(\vec{B}\) produced by current (moving charges) \(I\).
  • Interaction between \(\vec{E}\) and \(\vec{B}\) via rate of change of flux \(\Phi\) of one through loop of the other.

In integral form:

\[ \begin{align*} \oiint \vec{E} \cdot d{\vec{A}} &= \frac{1}{\epsilon_0} q & \oint \vec{E} \cdot d{\vec{l}} &= - \frac{d{\Phi_B}}{d{t}} \\ \oiint \vec{B} \cdot d{\vec{A}} &= 0 & \oint \vec{B} \cdot d{\vec{l}} &= \mu_0 \left(I + \epsilon_0 \frac{d{\Phi_E}}{d{t}} \right) \\ \end{align*} \]

Lorentz force law

Force \(\vec{F}\) on a particle with charge \(q\) and velocity \(\vec{v}\) by electric and magnetic fields \(\vec{E}\) and \(\vec{B}\).

\[ \vec{F} = q (\vec{E} + \vec{v} \times \vec{B}) \]

Coulomb’s law

Special case for stationary point charges where production and effect are expressed in one equation: force \(\vec{F}_1\) on particle with charge \(q_1\) due to electric field produced by particle with charge \(q_2\), where \(|r_{1,2}|\) is distance between particle \(1\) and \(2\) and \(\hat{r}_{1,2}\) is direction to \(1\) from \(2\).

\[ \vec{F}_1 = q_1 \frac{1}{\epsilon_0} q_2 \frac{1}{4\pi|r_{1,2}|^2} \hat{r}_{1,2} = \frac{q_1 q_2}{4\pi\epsilon_0} \frac{\hat{r}_{1,2}}{|r_{1,2}|^2} \]

References