• ⚡ In dielectrics there is a bound charge $\rho_\text{bound}=-\vec \nabla \cdot \vec P$ and a surface charge density $\sigma_\text{surface}=\vec P \cdot \hat n$.
• 🧲 In a magnet there is an equivalent bound current $\vec j_\text{bound}=\vec \nabla \times \vec M$ and a surface current $\vec j_\text{surface}=\vec M \times \hat n$ where $\hat n$ is a unit vector normal to the surface.

🗒️ Note: The polarization goes over to magnetization and the dot products become cross products

• The equations of magnetostatics are $\vec \nabla \cdot \vec B =0, \; \vec \nabla \times \vec B=\mu_0 \vec j$.

• For dielectrics we will split the current density $\vec j$ into contributions due to bound and free currents

  $$ \vec j=\vec j_\text{bound}+\vec j_\text{free} $$

  and Amperes law now becomes

  $$ \vec \nabla \times \vec B=\mu_0 (\vec j_\text{bound}+\vec j_\text{free})=\mu_0( \vec \nabla \times \vec M +\vec j_\text{free}) $$

• we now define the magnetic intensity vector

  $$ \vec H \equiv \frac{1}{\mu_0} \vec B - \vec M $$

  so that Amperes laws reads

  $$ \vec \nabla \times \vec H =\vec j_\text{free} $$

  where $\vec H$ is in [$\rm{Am}^{-1}$].

• The integral form of Amperes law becomes

  $$ \oint_\ell \vec H \cdot \text d \vec l=I_\text{free} $$

  🗒️ Note: the free current $I_\text{free}$ is the parameter we can control from outisde the material, just as $\rho_\text{free}$ was in the case of the dielectrics

• In diamagnetic and paramagnetic substances the magnetization is linked linearly to the magnetic field via

  $$ M=\frac{\chi_B}{\mu_0} \vec B $$

  which we can sub into the definition of the magnetic intensity vector to get

  $$ \vec H=\frac{1-\chi_B}{\mu_0}\vec B $$

• The magnetic energy becomes

  $$ U=\frac 12 \int \vec B \cdot \vec H \,\text d V $$

💼 Case: Consider the solenoid of length $\ell$

• Ampere’s law

  $$ \oint \vec H \cdot \text d\vec \ell =H\ell =N \ell I $$

  Hence

  $$ H=NI $$

• Using $\vec H=\frac{1-\chi_B}{\mu_0}\vec B$ we get

  $$ \mathcal E =-\frac{\mu_) N^2 \pi r^2 \ell}{1-\chi_B} \frac{\text dI}{\text dt} $$

  and the inductance

  $$ \mathcal L=-\frac{\mathcal E}{\dot I}=\frac{\mathcal L_\text{vac}}{1-\chi_B} $$

• Which makes us the define the relative permeability as

  $$ \mu_r\equiv \frac{1}{1-\chi_B} $$

  The magnetic field and magnetic intensity vectors are thus linked via

  $$ \vec B= \mu_r \mu_0 \vec H $$

Interference between magnets

Consider the boundary between two regions having relative permeability $\mu_r^{(1)},\mu_r^{(2)}$ with no free currents on the boundary. Hence one can find that

$$ \begin{aligned} \text{Ampere:}\quad \oint\vec H \cdot \text d \vec \ell &=0 \quad \Rightarrow \quad H_\parallel \text{ continuous} \\ \text{No monopoles:}\quad \int\vec B \cdot \text d \vec A &=0 \quad \Rightarrow \quad B_\perp \text{ continuous}

\end{aligned} $$

Ferromagnetism

When deriving paramagnetism we assumed that the intrinsic dipole moments didn’t interact with each other which enabled us to write down

$$ \vec M=\frac{nm^2_\text{intrinsic}}{3k_BT} \vec B_\text{ext} $$

These interactions create domains of size 0.1-1mm where in a given domain all magnetic moments have the same alignment. This is called ferromagnetism.