PX284 - Q2 - magnetization

bound charges can also flow and cause currents

the magnetic dipole moment:

\vec{m} = I \vec{A}

where, $\vec{A}$ is normal to surface following the right hand rule

units: A m $^{2}$
magnetization magnetic dipole moment per unit volume:

\vec{M} = \frac{\sum_{i \in V} {\vec{m}}_{i}}{V}

considering a bar magnet
the equivalent net cur rent will be a current around the outside of the material

uniform magnetization can be modelled as a uniform grid of current loops in x-y plane

considering a block of volume $Δ x Δ y Δ z :$

m_{z} = I Δ x Δ y

the current density:

j_{z} = \frac{I}{Δ z}

which is the current in the x-y plane

the magnetization:

M_{z} = \frac{m_{z}}{Δ x Δ y Δ z} = \frac{I}{Δ z} = j_{z}

generalize to all directions:

{\vec{j}}_{M} = \vec{M} \times \hat{n}

the net current between blocks centred at $x_{0}$ and $x_{0} + δ x :$

Δ \vec{I} = - \frac{\partial I}{\partial x} |_{x_{0}} Δ x \hat{y} = - \frac{\partial}{\partial x} M_{z} Δ x Δ z \hat{y}

the bulk current density (current per unit area):

J_{y} = - \frac{\partial}{\partial x} M_{z}

generalizing in 3D:

\vec{J} = \vec{\nabla} \times \vec{M}

units: A m $^{- 2}$