Magnetism – Advanced Study Notes
(For the TS EAMCET Physics Examination – MIT‑Level Treatment)
Notation
• 𝐁 – magnetic flux density (T)
• 𝐇 – magnetic field strength (A m⁻¹)
• 𝑀 – magnetization (A m⁻¹)
• 𝐽 – current density (A m⁻²)
• μ₀ – permeability of free space (4π × 10⁻⁷ H m⁻¹)
• μ – relative permeability (dimensionless)
• 𝐞 – unit vector
1. Electromagnetic Foundations
1.1 Maxwell’s Equations (in differential form)
| Equation | Physical Meaning | SI Units |
|---|---|---|
| ∇·𝐁 = 0 | No magnetic monopoles | T m⁻¹ |
| ∇·𝐇 = ρₘ | Magnetic charge density (ρₘ ≡ 0) | A m⁻² |
| ∇×𝐄 = –∂𝐁/∂t | Faraday’s law | V m⁻¹ s⁻¹ |
| ∇×𝐇 = 𝐉 + ∂𝐄/∂t | Ampère–Maxwell law | A m⁻² |
Key Insight: The first equation enforces that magnetic field lines are closed loops; the fourth equation shows that steady currents (∇×𝐇 = 𝐉) generate magnetic fields.
1.2 Constitutive Relations
For linear, isotropic media:
[
\mathbf{B} = \mu_0 \mathbf{H} + \mu_0 \mathbf{M}
\quad\text{or}\quad
\mathbf{B} = \mu \mu_0 \mathbf{H},
]
where (\mathbf{M}) is the magnetization (A m⁻¹).
In vacuum, (\mathbf{M}=0) and (\mu=\mu_0).
2. Magnetostatics – From First Principles
2.1 Biot–Savart Law
For a steady current element (I,d\boldsymbol{\ell}) located at (\mathbf{r}'), the magnetic field at (\mathbf{r}) is
[ d\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}, \frac{I,d\boldsymbol{\ell}\times \mathbf{\hat{r}}}{r^2}, \quad \mathbf{r}=\mathbf{r}-\mathbf{r}',; r=|\mathbf{r}|,; \mathbf{\hat{r}}=\frac{\mathbf{r}}{r}. ]
Derivation Sketch
- Start from the vector potential (\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|},d^3r').
- Use (\mathbf{B} = \nabla \times \mathbf{A}).
- Apply Stokes’ theorem to reduce volume integral to a line integral.
2.2 Ampère’s Law (Path Integral Form)
For a closed loop (\mathcal{C}):
[ \oint_{\mathcal{C}} \mathbf{H}\cdot d\boldsymbol{\ell} = I_{\text{enc}} + \frac{d}{dt}\int_{\mathcal{S}}\mathbf{D}\cdot d\mathbf{S}, ]
where (\mathcal{S}) is any surface bounded by (\mathcal{C}).
For static fields, the displacement current term vanishes.
Edge Case – Non‑Simply Connected Media
If the medium contains holes or topological constraints, the integral depends on the chosen surface; this is the essence of the Aharonov–Bohm effect in quantum mechanics.
3. Magnetic Fields of Simple Geometries
| Geometry | Field Inside | Field Outside | Formula |
|---|---|---|---|
| Infinite straight wire | (\displaystyle B = \frac{\mu_0 I}{2\pi r}) | (0) | |
| Solenoid (long, tightly wound) | (\displaystyle B = \mu_0 n I) | (0) | (n) = turns/m |
| Toroid (ideal) | (\displaystyle B = \frac{\mu_0 N I}{2\pi r}) | (0) | (N) = number of turns |
| Spherical shell of current | (\displaystyle B = \frac{\mu_0 I}{2r^2}) (inside) | (0) |
Note: For a finite solenoid, fringe fields at the ends are non‑zero. The exact solution requires elliptic integr