Moving Charges & Magnetism

Force on a point charge q moving with velocity v in field B. Always perpendicular to v — magnetic force does no work and cannot change kinetic energy.

Scalar form where θ is the angle between v and B. Maximum when v ⊥ B; zero when v ∥ B.

Macroscopic Lorentz force on a straight conductor of length L carrying current I in field B. Derived by summing qv × B over all free charges in the volume.

When v ⊥ B, magnetic force supplies centripetal force. Radius is proportional to momentum mv and inversely proportional to charge and field strength.

Period is independent of speed — a particle moving faster traces a larger circle in the same time. This independence is the principle underlying the cyclotron.

Natural frequency of circular orbit in B. The resonance condition in a cyclotron requires the alternating field frequency to match ν_c. Speed-independent up to relativistic effects.

When v has a component parallel to B, that component is unaffected by B. The particle traces a helix: radius r = mv⊥/qB from the perpendicular component, pitch p from the parallel component.

Maximum kinetic energy of a particle accelerated in a cyclotron of dee radius R. Obtained by substituting v = qBR/m into ½mv².

Magnetic field contributed by a current element Idl at position vector r from the element. The cross product encodes both magnitude (sinθ) and direction. Integrate over a complete circuit for total B.

Field at perpendicular distance d from a finite wire. φ₁ and φ₂ are angles subtended at the field point from the perpendicular foot to each end of the wire.

Limiting case of the finite wire formula (φ₁ = φ₂ = 90°). Also derived directly via Ampere's law. Field lines are concentric circles; direction by right-hand thumb rule.

Field at the centre of a circular arc of radius R subtending angle θ (radians) at the centre. For a complete loop (θ = 2π): B = μ₀I/2R.

Special case of the arc formula for θ = 2π. For N turns: B = μ₀NI/2R. Direction along the axis given by the right-hand curl rule.

Field at axial distance x from centre of a loop of radius R. At x = 0 reduces to μ₀I/2R. For x ≫ R falls as μ₀IR²/2x³ — the magnetic dipole field.

Line integral of B around any closed Amperian loop equals μ₀ times net enclosed current. Valid for steady currents. Most useful for high-symmetry current distributions.

n = N/L is turns per unit length. Derived via Ampere's law on a rectangular loop. Field is uniform inside, zero outside an ideal infinite solenoid.

N total turns, r = distance from toroid axis. Field is confined entirely inside the toroid body; B = 0 in the central cavity and outside. Follows from Ampere's law on concentric circular loops.

Two parallel wires separated by d attract for same-direction currents, repel for opposite. This result defines the SI ampere (pre-2019 definition).

Magnetic dipole moment for a planar loop of N turns, area A, carrying current I. Direction n̂ along the axis by right-hand rule. SI unit: A·m².

Torque tends to align the magnetic moment with B. θ is the angle between m and B. Maximum when m ⊥ B, zero when aligned. Basis of galvanometer and electric motor operation.

Minimum energy (stable equilibrium) at θ = 0; maximum (unstable) at θ = π. Change in PE between orientations equals work done against the torque.

α is deflection angle; k is torsional constant of the spring. At equilibrium, magnetic torque equals restoring torque. Deflection is proportional to current — galvanometer is a linear instrument.

Deflection per unit current. Increased by larger N, B, A or weaker spring k. Voltage sensitivity = S_I / G where G is galvanometer resistance.

Low resistance S in parallel with galvanometer of resistance G allows current (I − Ig) to bypass, giving full-scale deflection at current I. Converted ammeter has very low net resistance.

High resistance R in series gives full-scale deflection at terminal voltage V. Very high total resistance minimises loading on the circuit under measurement.