Short on time? This page is the recap. For Figure 1, the derivations and the worked examples, use the full study guide. This guide is for the calculus-based course.
Recap
- B is a vector field. It gives the magnetic force on moving charges, currents and magnetic materials. Unit: tesla (T). A charge at rest feels no magnetic force.
- Sources of B are dipoles or combinations of dipoles, never monopoles. Each dipole has an N and an S pole. Cut a magnet and each piece is a dipole.
- Like poles repel; unlike poles attract. A compass (a small dipole) turns so that its N end points along B.
- A dipole’s field gets weaker with distance from it.
- Dipoles come from the circular or spinning motion of charge, mainly electrons. Permanent and induced magnetism both come from aligned dipoles.
- Earth’s field is modelled as a dipole field. The pole near the geographic North Pole is a magnetic south pole.
- μ₀ = 4π × 10⁻⁷ T·m/A is the constant permeability of free space. A material’s μ depends on its composition and changes with temperature, orientation and field strength.
Key relationships
| Idea | Statement |
|---|---|
| Gauss’s law for magnetism (Maxwell’s second equation) | ∮B·dA = 0 for any closed surface |
| Flux through an open surface | Φ_B = ∫B·dA; flat area in uniform B: Φ_B = BA cos θ |
| Unit of flux | 1 Wb = 1 T·m² |
| Missing flux | Φ_last part = −(sum of fluxes through all other parts) |
| Radial field near an axis (symmetric field) | B_r = −(r/2) dB_z/dz |
| Material types | ferro: μ ≫ μ₀; para: μ slightly > μ₀; dia: μ slightly < μ₀ |
Assumptions
- Outward area vectors on closed surfaces, as for the electric Gauss’s law.
- “Uniform field” means the same B at every point of the surface.
- The radial-field result needs symmetry about the z-axis and a point close to the axis.
Mistakes to avoid
- Ending field lines at the S pole. They carry on through the magnet.
- Non-zero net flux around a pole. Every closed surface has zero net magnetic flux.
- Forgetting the sign. Flux is negative where lines go in through a closed surface.
- Calling diamagnetism rare. All materials have it; it is just weak.
- Treating μ as a fixed material constant. Only μ₀ is fixed.
- Earth’s poles. The compass N end points towards a magnetic S pole.
Quick self-check
- A closed surface has 3.0 × 10⁻⁴ Wb entering through its base and 1.8 × 10⁻⁴ Wb leaving through its top. What is the flux through the rest? (+1.2 × 10⁻⁴ Wb, leaving)
- An unmagnetised steel paper clip is attracted to either pole of a magnet. Why? (The field induces aligned dipoles in it, with the near end opposite to the nearby pole)
- Aluminium is weakly attracted to a strong magnet but keeps no magnetism afterwards. Which type is it? (Paramagnetic)
Next: practice questions.