Magnetism – Study Notes for PSU‑Jobs (GATE)
Topic: Magnetism
Course: Physics – Electromagnetism
Exam: PSU‑Jobs (GATE)
Time to read: ~30 min
Time to revise: 10 min
1. Simple Explanation
- Magnetic fields are produced by moving electric charges (currents) or by changing electric fields.
- They exert forces on moving charges or on magnetic dipoles.
- The magnetic field vector B points in the direction a north magnetic pole would be pushed; its magnitude is measured in tesla (T).
- Flux (Φ) is the total magnetic field passing through a surface: Φ = ∫𝐵·d𝐬.
- Magnetization (M) describes how a material becomes a magnetic dipole when placed in an external field.
2. Key Points & Equations
| Concept | Symbol | Formula | Notes |
|---|---|---|---|
| Magnetic field of a moving point charge | – | 𝐵 = (μ₀/4π) (q 𝐯 × 𝐫̂) / r² | Right‑hand rule: 𝐯 cross 𝐫̂ |
| Biot–Savart Law | – | d𝐵 = (μ₀/4π) I (d𝐥 × 𝐫̂)/r² | For a current element; integrate for wire/shapes |
| Magnetic field of a long straight wire | – | B = μ₀I / (2πr) | Circular field lines |
| Magnetic field of a circular loop | – | B = (μ₀I a²) / [2(a² + z²)^(3/2)] | Along axis; at centre → B = μ₀I/(2a) |
| Magnetic field of a solenoid | – | B = μ₀ n I | n = turns per unit length |
| Magnetic field of a toroid | – | B = (μ₀ N I) / (2πr) | r = radial distance from core |
| Ampère’s Law | – | ∮𝐵·dl = μ₀ I_enc | For high‑symmetry cases |
| Magnetic Flux | Φ | ∫𝐵·d𝐬 | Units: Wb (weber) |
| Faraday’s Law (induction) | – | ℰ = - dΦ/dt | EMF induced in a circuit |
| Lenz’s Law | – | “Opposite” tendency | Sign of emf |
| Magnetic Dipole Moment | μ | μ = I A (for a loop) | Direction: right‑hand rule |
| Torque on a dipole | τ | τ = μ × B | Magnitude τ = μ B sinθ |
| Magnetic Potential Energy | U | U = - μ·B | Minimum when μ‖B |
| Magnetization | M | M = (magnetic dipole moment per unit volume) | B = μ₀(H + M) |
| Magnetic Permeability | μ | μ = μ₀ μ_r | μ_r = relative permeability |
| Magnetic Susceptibility | χ_m | M = χ_m H | χ_m > 0 (paramagnetic), <0 (diamagnetic) |
| Coil Inductance | L | Φ = L I | L depends on geometry & μ |
Magnetic Materials
| Material | χ_m | μ_r | Behaviour |
|---|---|---|---|
| Ferromagnetic (Fe, Ni, Co) | > 1000 | > 1000 | Strong attraction |
| Paramagnetic | 10⁻³–10⁻² | 1–1.1 | Weak attraction |
| Diamagnetic | <0 | <1 | Weak repulsion |
| Superconductor | –∞ | 0 | Expels B (Meissner effect) |
3. Worked Example
Problem (GATE‑style)
A thin circular loop of radius (a = 10\text{ cm}) carries a current (I = 5\text{ A}). A second identical loop is placed coaxially at a distance (z = 20\text{ cm}) along the axis. Find the magnetic force between the two loops.
Step‑by‑step Solution
- Magnetic Field on the Axis of a Loop
For a single loop at a point on its axis: [ B(z) = \frac{\mu_0 I a^2}{2(a^2+z^2)^{3/2}} ] Insert values: [ a = 0.10\text{ m}, ; z = 0.20\text{ m} ] [ B = \frac{(4\pi \times10^{-7}) (5)(0.10)^2} {2\big((0.10)^2+(0.20)^2\big)^{3/2}} ] Compute denominator: [ (0.10)^2+(0.20)^2 = 0.01 + 0.04 = 0.05 ] [ (0.05)^{3/2} = (0.05)^{1.5} \approx 0.01118 ] Numerator: [ 4\pi \times10^{-7} \times 5 \times 0.01 = 2\pi \times10^{-7} \approx 6.283\times10^{-7} ] Thus, [ B \approx \frac{6.283\times10^{-7}}{2 \times 0.01118} \approx \frac{6.283\times10^{-7}}{0.02236} \approx 2.81\times10^{-5},\text{T