Short on time? This page is the recap. For the derivations, the figures and the worked examples, use the full study guide. This topic is in the calculus-based course.
Recap
- Ampère’s law links the line integral of B round a closed path (an Amperian loop) to the net current through it.
- Sign rule: curl your right fingers along the direction of travel round the loop; current along your thumb is positive.
- The law is always true, but it gives B directly only when symmetry makes B constant and parallel to parts of the path, and perpendicular (or zero) on the rest.
- The course applies it quantitatively to long straight wires, long solenoids, and conducting slabs or cylinders carrying a current density.
- Treat solenoids as ideal unless told otherwise: uniform field inside, negligible field outside.
- μ₀ = 4π × 10⁻⁷ T·m/A; μ₀/(2π) = 2 × 10⁻⁷ T·m/A.
Key relationships
| Current distribution | Amperian loop | Field |
|---|---|---|
| Long wire, or outside any long cylinder | coaxial circle | B = μ₀I/(2πr) |
| Inside a uniform solid cylinder (r < R) | coaxial circle | B = μ₀Ir/(2πR²) |
| Non-uniform J(r) in a cylinder | coaxial circle | I_enc = ∫₀ʳ J(r′)2πr′ dr′, then B = μ₀I_enc/(2πr) |
| Inside an empty tube, or outside a coaxial cable | coaxial circle | B = 0 |
| Slab, thickness t, inside at distance z from the mid-plane | rectangle | B = μ₀Jz |
| Slab, outside | rectangle | B = μ₀Jt/2, independent of distance |
| Long solenoid, n = N/L | rectangle | B = μ₀nI inside, ≈ 0 outside |
| Maxwell’s fourth equation | any | ∮B·dℓ = μ₀I_enc + μ₀ε₀ dΦ_E/dt (idea only) |
Assumptions behind the results
- Steady currents, and exact symmetry: “infinite” wires, slabs and solenoids.
- For the uniform-cylinder results, current is spread evenly over the cross-section.
- No magnetic materials nearby; the space has permeability μ₀.
Mistakes to avoid
- Total I inside a conductor. Use I_enc(r).
- J × area when J varies. Integrate J 2πr dr.
- ∮B·dℓ = 0 ⇒ B = 0. Wrong: it means zero net enclosed current.
- Ignoring the sign rule. Opposite currents through the loop subtract.
- N instead of n. Use turns per metre.
- Ampère’s law on a single loop or short wire. No usable symmetry; use Biot-Savart.
Quick self-check
- An Amperian loop encloses 5.0 A one way and 2.0 A the other. What is ∮B·dℓ? (μ₀ × 3.0 A = 3.8 × 10⁻⁶ T·m)
- Inside a uniform solid wire, how does B change if r is halved? (It halves, since B ∝ r)
- A solenoid has 1000 turns per metre and carries 2.0 A. What is B inside? (2.5 × 10⁻³ T)
Next: practice questions.