How it works
The Biot-Savart law gives the differential magnetic field contribution dH = I(dl × r̂)/(4πr²), used for arbitrary current geometries. For a finite straight wire of length L at perpendicular distance d, H = I(sinα₂ − sinα₁)/(4πd). Ampere's circuital law in integral form: ∮H·dl = I_enclosed. For an infinite straight wire: H = I/(2πρ) φ̂ in cylindrical coordinates. A toroidal coil with N turns and mean radius R has H = NI/(2πR) inside the core, zero outside — a key result for transformer analysis.
Key points to remember
Magnetic flux Φ = ∫B·dS, measured in Webers. The boundary conditions: normal B is continuous across interfaces (B₁ₙ = B₂ₙ); tangential H is discontinuous by the surface current density (H₁ₜ − H₂ₜ = K). Magnetic vector potential A is defined by B = ∇×A, and satisfies Poisson's equation ∇²A = −µJ. For a solenoid with n turns/meter carrying current I: B = µ₀µr·nI inside, zero outside. Energy stored in magnetic field: W_m = ½µ₀∫|H|² dV. Inductance L = NΦ/I, and mutual inductance M = N₂Φ₁₂/I₁.
Exam tip
The examiner always asks you to find H inside and outside a coaxial cable or toroid using Ampere's law — define your Amperian loop clearly, calculate enclosed current for each region, and present results in a piecewise expression.