# Magnetic Materials

Physics II · Electromagnetism · https://tryals.app/learn/physics-ii/magnetic-materials

## Matter Responds to B

Just as a dielectric polarises, matter **magnetises**. The magnetisation $\mathbf{M}$ is the magnetic moment per unit volume, and it is equivalent to **bound current densities**:

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

These are real currents, circulating atomic currents that do not transport charge through the material.

To keep Ampère's law usable, magnetostatics defines the **auxiliary field**

$$\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}$$

whose curl counts only **free** current: $\nabla \times \mathbf{H} = \mathbf{J}_f$. This is the exact counterpart of $\mathbf{D}$ in electrostatics, and it inherits the same warning, since $\mathbf{B}$ is what exerts force.

For linear media $\mathbf{M} = \chi_m \mathbf{H}$ and $\mathbf{B} = \mu_0\mu_r\mathbf{H}$ with $\mu_r = 1 + \chi_m$. Here the parallel with dielectrics breaks in an important way: $\chi_m$ can be **negative**.

| Class | $\chi_m$ | Behaviour |
|---|---|---|
| Diamagnetic | Small negative | Weakly repelled; field slightly reduced |
| Paramagnetic | Small positive | Weakly attracted; field slightly raised |
| Ferromagnetic | Large, non-linear | Strongly attracted; field hugely raised |

Diamagnetism is universal, an induced response opposing the change, present in every material but usually masked. Paramagnetism comes from permanent moments partially aligning against thermal disorder, following **Curie's law** $\chi_m \propto 1/T$.

Ferromagnets have moments aligned in **domains** by exchange coupling. They are non-linear and history-dependent: the $B$-$H$ curve forms a **hysteresis loop**, whose area is the energy dissipated per cycle. Above the **Curie temperature** thermal motion destroys the ordering and the material becomes paramagnetic.

> **Common pitfall:** assuming $\mu_r > 1$ because $\varepsilon_r > 1$ always is. Diamagnets have $\mu_r$ slightly *below* 1 and are pushed out of a field, the reason a superconductor, the perfect diamagnet with $\chi_m = -1$, levitates.

## Practice questions

9 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. A material has magnetic susceptibility $\chi_m = -2.0 \times 10^{-5}$. Set its relative permeability on the scale, which runs from 0.9999 to 1.0001 in steps of 0.00001.

**Answer:** 0.99998 (within ±0.000015)

**Why:** $\mu_r = 1 + \chi_m = 1 - 2.0\times10^{-5} = 0.99998$, just *below* 1, which is the diamagnetic signature and has no dielectric counterpart. This is the one place the electric and magnetic parallels genuinely diverge.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/a-material-has-magnetic-susceptibility-m-2-0-10-set-its

### 2. The relative permeability of a diamagnetic material is slightly less than 1.

**Answer:** True

**Why:** True, diamagnets have $\chi_m < 0$, giving $\mu_r$ just below 1. This is where the magnetic case departs from the dielectric one, where $\varepsilon_r$ is always above 1.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/the-relative-permeability-of-a-diamagnetic-material-is-slightly-less

### 3. Why does magnetostatics introduce the auxiliary field $\mathbf{H}$?

A. It represents the fundamental force field acting on moving free charges
B. Its divergence vanishes everywhere, ensuring magnetic monopoles cannot exist
C. It isolates bound surface currents so volume currents can be neglected
D. Its curl counts only free current, so Ampère’s law stays usable in matter

**Answer:** D. Its curl counts only free current, so Ampère’s law stays usable in matter

**Why:** $\nabla\times\mathbf{H} = \mathbf{J}_f$ hides the bound magnetisation currents, so Ampère’s law can be applied with only the current you actually drive. Force on a moving charge is still $q\mathbf{v}\times\mathbf{B}$.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/why-does-magnetostatics-introduce-the-auxiliary-field-h

### 4. Bound currents in magnetised matter are real microscopic currents, yet they transport no net charge across macroscopic distances. Why does this distinction prevent bound current from appearing in the curl of the auxiliary field H?

A. Auxiliary fields track only controllable external flows
B. It produces no macroscopic magnetic field in the bulk
C. Bound current only exists on the outer surface boundaries
D. Closed microscopic orbits do not satisfy charge continuity

**Answer:** A. Auxiliary fields track only controllable external flows

**Why:** Microscopic bound currents generate genuine B-fields and obey continuity, but they cannot be routed or metered externally. Defining H isolates the engineer's transport currents from material response.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/bound-currents-in-magnetised-matter-are-real-microscopic-currents

### 5. Which statements about ferromagnetism are correct?

A. Its response to an applied field is non-linear
B. Its susceptibility is small and negative
C. The material contains domains of aligned moments
D. It shows hysteresis, so its state depends on its history

**Answer:** A. Its response to an applied field is non-linear; C. The material contains domains of aligned moments; D. It shows hysteresis, so its state depends on its history

**Why:** Domains, non-linearity and history dependence are the ferromagnetic signature. A small negative susceptibility is diamagnetism, the opposite extreme.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/which-statements-about-ferromagnetism-are-correct

### 6. Match each magnetic quantity to what it describes.

**Answer:**

- Magnetisation $\mathbf{M}$ → Magnetic moment per unit volume
- Auxiliary field $\mathbf{H}$ → A field whose curl counts free current only
- Curie temperature → Where a ferromagnet becomes paramagnetic
- Hysteresis loop area → Energy dissipated per magnetisation cycle

**Why:** $\mathbf{M}$ measures the aligned moments, $\mathbf{H}$ keeps Ampère’s law usable, the Curie temperature is where ordering collapses, and the loop area is the energy paid per cycle.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/match-each-magnetic-quantity-to-what-it-describes

### 7. Sort each statement by whether it applies to a dielectric or a magnetic material.

**Answer:**

- Dielectrics only: Relative constant is > 1
- Magnetic materials only: Relative constant is < 1, Retains state after field
- Both: Auxiliary field hides bounds, Unit volume susceptibility

**Why:** The parallel is close, both define an auxiliary field and a susceptibility. It breaks in two places: magnetic susceptibility can be negative, and ferromagnets remember their history in a way no dielectric does.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/sort-each-statement-by-whether-it-applies-to-a-dielectric-or-a

### 8. Diamagnetism occurs only in materials that have no permanent magnetic moments.

**Answer:** False

**Why:** False, diamagnetism is a universal *induced* response opposing any change in field, present in every material including strongly paramagnetic ones. It is simply masked when stronger effects are also present, not absent.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/diamagnetism-occurs-only-in-materials-that-have-no-permanent-magnetic

### 9. Complete the description of how the two auxiliary fields parallel one another.

**Answer:** In a dielectric the displacement field hides the **bound** charge so that Gauss’s law counts only free charge. In a magnetic material the **auxiliary H** field plays the same role, hiding the bound **currents** so that Ampère’s law counts only free **current**.

**Why:** $\mathbf{D}$ removes bound charge from Gauss’s law and $\mathbf{H}$ removes bound current from Ampère’s law. Both are bookkeeping devices, and in both cases the force is still exerted by the original field.

Page: https://tryals.app/practice/physics-ii/magnetic-materials/complete-the-description-of-how-the-two-auxiliary-fields-parallel-one