Physics Notes: Angular Momentum, Magnetic and Spin

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