AP Physics C: E&M Unit 4 Notes — Understanding Magnetic Forces

Magnetic force (on a charge)

The force a magnetic field exerts on a moving electric charge; it is zero for a stationary charge.

Lorentz force

The total electromagnetic force on a charge: F⃗ = qE⃗ + q(v⃗ × B⃗).

Magnetic part of the Lorentz force

The magnetic force on a moving charge: F⃗_B = q(v⃗ × B⃗).

Cross product (×)

A vector operation whose result is perpendicular to both vectors; for magnetic force it makes F⃗_B perpendicular to v⃗ and B⃗.

Magnetic force magnitude

F_B = |q|vB sinθ, where θ is the angle between v⃗ and B⃗.

Angle dependence (sinθ factor)

Magnetic force depends on the sine of the angle between motion and field; parallel/antiparallel gives zero force, perpendicular gives maximum force.

Right-hand rule for v⃗ × B⃗

For a positive charge: point fingers along v⃗, curl toward B⃗, thumb gives F⃗_B direction.

Negative charge direction reversal

If q is negative (e.g., an electron), the magnetic force direction is opposite the right-hand-rule result for a positive charge.

Magnetic forces do no work

Because F⃗B is perpendicular to velocity (and displacement), WB = 0, so kinetic energy and speed stay constant in a purely magnetic field.

Tesla (T)

SI unit of magnetic field strength; 1 T = 1 N/(C·m/s) based on F_B = |q|vB sinθ.

Current

The flow of electric charge; a current is many moving charges and can experience magnetic forces in a field.

Force on a current-carrying wire segment

Vector form: F⃗ = I(L⃗ × B⃗), where L⃗ points in the direction of conventional current.

Force magnitude on a straight wire segment

F = ILB sinθ, where θ is the angle between the wire/current direction and B⃗.

Conventional current direction

The direction positive charges would move; used for L⃗ in F⃗ = I(L⃗ × B⃗), even though electrons drift opposite in metals.

Drift velocity

The average velocity of charge carriers in a conductor; microscopic picture connecting q(v⃗ × B⃗) to the macroscopic wire force.

Force per unit length between parallel wires

For two long parallel wires: (F/L) = (μ0 I1 I2)/(2πr).

Permeability of free space (μ0)

Constant in the parallel-wire force law; μ0 = 4π × 10^-7 N/A^2.

Parallel currents attract/repel rule

Two parallel wires with currents in the same direction attract; currents in opposite directions repel.

Velocity components relative to B⃗

Decompose v⃗ into v⃗∥ (parallel to B⃗, no magnetic force) and v⃗⊥ (perpendicular to B⃗, causes deflection).

Centripetal force role of magnetic force

When v⃗ ⊥ B⃗, magnetic force acts as the centripetal force, changing direction of motion without changing speed.

Gyroradius (radius of circular motion)

For v ⊥ B: r = mv/(|q|B); larger m or v increases r, larger |q| or B decreases r.

Cyclotron angular frequency (ω)

Angular speed of circular motion in a uniform magnetic field: ω = |q|B/m (nonrelativistic).

Cyclotron period (T)

Time for one revolution: T = 2πm/(|q|B); independent of particle speed (nonrelativistic).

Helical motion

Motion when both v∥ and v⊥ are present: circular motion from v⊥ combined with constant translation along B⃗ from v∥.

Velocity selector

Crossed-field device where no deflection requires qE = qvB, selecting speed v = E/B (charge cancels).