The Lorentz Force Law

For moving point charge…

\[F_{mag} = Q(\vec{v} \times \vec{B})\]

For current….

\[F_{mag} = I \int d\vec{l} \times \vec{B}\]

The Biot-Savart Law:

\[\vec{B}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{\vec{I} \times \hat{\mathscr{r}}}{\mathscr{r}^2}dl^\prime\]

\(\mu_0\) is the permeability of free space. \(\mathscr{r}\) is the vector from the source point to the point \(\vec{r}\).

For a steady current…

\[\vec{B}(\vec{r}) = \frac{\mu_0}{4\pi}I \int \frac{\vec{dl^\prime} \times \hat{\mathscr{r}}}{\mathscr{r}^2}\]

For a straight long current…

\[B = \frac{\mu_0 I}{2\pi d}\]

You can find the direction of \(\vec{B}\) using the right hand rule.

Ampere’s Law (the curl of \(\vec{B}\))

Differential form:

\[\nabla \times \vec{B} = \mu_0 \vec{J}\]

Integral form:

\[\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}\]

or for a current density \(\vec{J}\):

\[\oint \vec{B}\cdot d\vec{l} = \int\vec{J}\cdot d\vec{a}\]

The Divergence of B

Unlike E the divergence of B is zero:

\[\nabla \cdot \vec{B} = 0\]

The Vector Potential:

\[\vec{B} = \nabla \times \vec{A}\]

(We can assume \(\vec{A}\) exists because the divergence of B is zero. The curl of a divergence is zero.)

Poisson’s equation for \(\vec{A}\):

\[\nabla^2 \vec{A} = -\mu_0 \vec{J}\]

Note this is really three equations, one for each component of \(\vec{A}\).

Boundary conditions:

For the magnetic field and its associated vector potential:

  • A is continuous
  • B perp is continuous!!
  • B parallel is discontinuous
\[B_{above}^\perp = B_{below}^\perp\] \[B_{above}^\| - B_{below}^\| = \mu_0 K\]

where \(\vec{K}\) is the surface current. (Current per area, so surface charge density times velocity).

Magnetic Dipole:

\[\vec{m} = I\vec{a}\]

where \(\vec{a}\) is the vector area of the loop. Figure out which was it points using the right-hand rule.

\[A_{dip}(\vec{r}) = \frac{\mu_0}{4\pi} \frac{\vec{m}\times\hat{r}}{r^2}\]