The Field of a Magnetized Object

In the previous section, we defined magnetization $\mathbf{M}$ as the magnetic dipole moment per unit volume. But how do we actually calculate the magnetic field $\mathbf{B}$ produced by an object with a given magnetization?

Just as polarization $\mathbf{P}$ effectively accumulates charge, magnetization $\mathbf{M}$ effectively creates currents. We call these Bound Currents.

1. Bound Currents

Suppose we have a chunk of magnetized material. The total vector potential $\mathbf{A}$ at an observation point is the sum of the potentials produced by all the infinitesimal dipoles in the volume.

Through mathematical transformation (integration by parts), we discover that the potential produced by magnetization is exactly the same as the potential produced by a specific volume current density $\mathbf{J}_b$ and a surface current density $\mathbf{K}_b$.

\mathbf{J}_b = \nabla \times \mathbf{M} $$ $$ \mathbf{K}_b = \mathbf{M} \times \mathbf{\hat{n}}

Here, $\mathbf{J}_b$ acts throughout the body of the material, while $\mathbf{K}_b$ flows along the surface (where $\mathbf{\hat{n}}$ is the surface normal).

⚡ Electrostatic Analogy

Recall that for a polarized object, we defined Bound Charges: $$ \rho_b = -\nabla \cdot \mathbf{P} \quad \text{and} \quad \sigma_b = \mathbf{P} \cdot \mathbf{\hat{n}} $$ Electric polarization acts like a pile-up of charge (divergence), whereas magnetization acts like a circulation of current (curl).

Deep Dive: Derivation & Example

2. Physical Interpretation

Are these "bound currents" real? Yes. They represent the cumulative effect of the microscopic atomic current loops.

Imagine a uniform grid of tiny current loops. Inside the material, the "up" current of one loop is canceled by the "down" current of its neighbor. There is no net current flow in the bulk.

However, at the very edge of the material, there is no neighbor to cancel the current. This results in a net current flowing around the perimeter—this is the Surface Current $\mathbf{K}_b$.

\text{Internal Cancellation} \rightarrow \mathbf{J}_b = 0 $$ $$ \text{Uncancelled Edge} \rightarrow \mathbf{K}_b \neq 0

If the magnetization is non-uniform (stronger in one region than another), the internal cancellation is incomplete, resulting in a net Volume Current $\mathbf{J}_b$.

Deep Dive: Physical Interpretation

3. The Magnetic Field Inside Matter

Once we determine the bound currents $\mathbf{J}_b$ and $\mathbf{K}_b$, the problem reduces to a standard magnetostatics problem. We can calculate the field $\mathbf{B}$ using Biot-Savart or Ampere’s Law, treating the bound currents exactly like "free" currents running through a wire.

The macroscopic field inside the material is the average field over a region large enough to smooth out the atomic variations but small enough to preserve the material's geometry.

\text{Treat } \mathbf{J}_b, \mathbf{K}_b \text{ as sources for } \mathbf{B}

For example, a uniformly magnetized sphere produces a field identical to a rotating charged shell (a surface current). Inside the sphere, the field is perfectly uniform!

⚡ Electrostatic Analogy

Electric: A uniformly polarized sphere produces a uniform electric field inside, opposing $\mathbf{P}$.
Magnetic: A uniformly magnetized sphere produces a uniform magnetic field inside, parallel to $\mathbf{M}$.