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Magnetostatics: Biot-Savart and Ampere's Law

Biot-Savart law, magnetic field of a wire, loop and solenoid, Ampere's circuital law, magnetic flux and force on conductors, magnetic materials, boundary conditions and inductance.

Steady currents produce magnetic fields. The field H (A/m) and flux density B = μH (tesla) describe them.

Biot–Savart law

dH = I dl × â_R/(4π R²)

Results:

SourceField
Infinite straight wireH = I/(2πr), circles around the wire
Finite wireH = (I/(4πr))(sin α₂ − sin α₁)
Centre of a circular loop (radius a)H = I/(2a)
On the axis of a loop at distance zH = I a²/(2(a² + z²)^(3/2))
Long solenoid (n turns/m)H = nI inside, ~0 outside
ToroidH = NI/(2πr) inside

Ampere's circuital law

∮ H·dl = I_enclosed; point form ∇ × H = J.

∇·B = 0: there are no magnetic monopoles; flux lines always close.

Forces

  • On a charge: F = q(E + v × B) (Lorentz force).
  • On a conductor: F = I L × B.
  • Between two parallel wires: force per metre = μ₀I₁I₂/(2πd); attractive for currents in the same direction.
  • μ₀ = 4π × 10⁻⁷ H/m.

Magnetic materials and boundaries

  • Diamagnetic (μ_r slightly < 1), paramagnetic (slightly > 1), ferromagnetic (μ_r ≫ 1, with hysteresis).
  • At a boundary (no surface current): normal B is continuous; tangential H is continuous.

Inductance

  • L = NΦ/I (flux linkage per ampere). Energy W = ½LI²; energy density ½μH².
  • Solenoid: L = μN²A/l. Toroid: L = μN²A/(2πr).
  • Coaxial cable (per metre, external): L = (μ/(2π)) ln(b/a).
  • Two-wire line (per metre): L ≈ (μ/π) ln(d/a).

Worked example

Two parallel wires 0.1 m apart carry 50 A each in the same direction.

  • F = 4π × 10⁻⁷ × 50 × 50/(2π × 0.1) = 5 mN per metre, attractive.

Practice

GATE-styleNAT · numerical answer

What is the magnetic field intensity (A/m) 0.5 m from a long wire carrying 10 A? (Give 3 decimal places.)

Show answer and explanationAnswer: 3.183

H = I/(2πr) = 10/π = 3.183 A/m.

GATE-styleNAT · numerical answer

A circular loop of radius 0.1 m carries 2 A. What is H (A/m) at its centre?

Show answer and explanationAnswer: 10

H = I/(2a) = 2/0.2 = 10 A/m.

GATE-styleNAT · numerical answer

A solenoid has 1000 turns over 0.5 m and carries 2 A. What is H inside (A/m)?

Show answer and explanationAnswer: 4000

H = nI = 2000 × 2 = 4000 A/m.

GATE-styleMCQ · one correct answer

Which Maxwell equation states that magnetic monopoles do not exist?

Show answer and explanationAnswer: B

Zero divergence of B means every flux line closes on itself.