Physics

Magnetostatics

Lorentz force, Biot–Savart, Ampère, and the solenoid.

Basics

Lorentz force

The magnetic force on a charge is q v × B. It is perpendicular to velocity, so it bends direction without changing speed. In uniform B a charge circles with radius mv/(qB). With an electric field too, F = q(E + v×B). Use the right-hand rule for direction.

Biot–Savart

A current element Idℓ makes dB falling as 1/r² with direction Idℓ × r̂. A long straight wire wraps B = μ₀ I /(2π r). At the centre of a loop, B = μ₀ I /(2R). A bad integration path misses the symmetry. Vacuum permeability μ₀ sits in front.

Ampère’s law

Around a closed loop, ∮ B·dℓ = μ₀ I_enc. Good symmetry hands you |B| at once. Inside a long ideal solenoid B≈μ₀ n I. A toroid wraps in the azimuthal direction. Maxwell–Ampère adds displacement current. For steady currents, conduction current is enough.

Wires and torque

A straight wire feels F = I ℓ × B. Parallel wires attract if currents match and repel if opposite. A loop has moment μ = I A n̂ and torque τ = μ × B that tries to align it. Motors and meters use that torque. Macroscopic EM assumes no magnetic monopoles.

Formulas

Lorentz force

F = q (E + v × B)

Magnetic part alone is q v × B.

Symbols

v velocity
B magnetic field

Long straight wire

B = μ₀ I / (2π r)

Perpendicular distance r from the wire. Circles by the right-hand rule.

Symbols

I current

Long solenoid (interior)

B ≈ μ₀ n I

n is turns per unit length.

Symbols

n turns per unit length

Cyclotron radius

r = m v_⊥ / (|q| B)

Speed component perpendicular to B.

Symbols

v_⊥ perpendicular speed

Key table

Tesla	Earth’s surface \|B\| is tens of μT; MRI is a few T
Old ampere	before 2019 the ampere was defined via force between parallel wires
∇·B=0	magnetic flux lines close; no monopoles (macro)