The note is summarized from several good textbook, includes Goldstein’s ‘Classical Mechanics’, Griffiths’s ‘Introduction to Electrodynamics’ and Cohen’s ‘Quantum Mechanics Volume 1’

Angular Momentum

The angular Momentum of a Particle is defined as: \(\vec{L}=\vec{r}\times \vec{p}\) The moment of force or torque is: \(\vec{N}=\vec{r}\times \vec{F}=\vec{r}\times\frac{d}{dt}(m\vec{v})=\frac{d}{dt}(\vec{r}\times m\vec{v})=\dot{L}\) Conservation Theorem for the Angular Momentum of a Partical$$: If the total torque, $\vec{N}$ is zero then $\dot{L}=0$, and the angular momentum $\vec{L}$ is conserved.

To a system of particles: \(\dot{L}=\sum_i\vec{r}_i\times \vec{F}_i=\sum_i \vec{r}_i\times\vec{F}_i^(e)+\sum_{i,j;i\neq j}\vec{r}_i\times\vec{F}_{ji}\) If all internel force can cancel each other, we have the conservation of Angular momentum of a system of particles: \(\frac{d\vec{L}}{dt}=\vec{N}^{(e)}\)

Magnetostatics

First we establish the magetic theory of charges, then we build up the model of matters.

magnetic forces $\vec{F}{mag}=Q(\vec{v}\times \vec{B})$ comes from experimental observation. magnetic forces do no work $\vec{F}{mag}\cdot d\vec{l}=\vec{F}_{mag}\cdot\vec{v}dt=0$

The Biot-Savart Law

Stationary charges produce electric fields that are constant in time. Here we assume there exist a steady currents that is also constant in time. Steady chage in electrostatics is expressed as $\partial \rho/\partial t=0$. Here we must have another: \(\nabla\cdot\vec{J}=0\)

Biot-Savart law \(\vec{B}(r)=\frac{\mu_0}{4\pi}\int\frac{\vec{I}\times\hat{\zeta}}{\zeta^2}dl'=\frac{\mu_0}{4\pi}I\int\frac{d\vec{l}'\times\hat{\zeta}}{\zeta^2}\)

Some examples are needed here, and the origin of this Biot-Savart law, why is it so?

Properties of $\vec{B}$

\(\nabla\times\vec{B}=\mu_0\vec{J}\) \(\nabla\cdot\vec{B}=0\) How to derive these