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: $\tilde{F} = q \tilde{E} + q(\tilde{v} \times \tilde{B})$.

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 $\tilde{F}_B$ perpendicular to $\tilde{v}$ and $\tilde{B}$.

Magnetic force magnitude

$F_B = |q|vB \theta$, where $\theta$ is the angle between $\tilde{v}$ and $\tilde{B}$.

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 $\tilde{v} \times \tilde{B}$

For a positive charge: point fingers along $\tilde{v}$, curl toward $\tilde{B}$, thumb gives $\tilde{F}_B$ 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 $\tilde{F}_B$ is perpendicular to velocity (and displacement), $W_B = 0$, so kinetic energy and speed stay constant in a purely magnetic field.

Tesla (T)

SI unit of magnetic field strength; $1 T = 1 \frac{N}{(C \bullet m/s)}$ based on $F_B = |q|vB \sin\theta$.

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: $\tilde{F} = I(\tilde{L} \times \tilde{B})$, where $\tilde{L}$ points in the direction of conventional current.

Force magnitude on a straight wire segment

$F = ILB \theta$, where $\theta$ is the angle between the wire/current direction and $\tilde{B}$.

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: $\frac{F}{L} = \frac{\tilde{\nu}_0 I_1 I_2}{2\theta r}$.

Permeability of free space ($\tilde{\nu}_0$)

Constant in the parallel-wire force law; $\tilde{\nu}_0 = 4\theta \times 10^{-7} N/A^2$.

Parallel currents attract/repel rule

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

Velocity components relative to $\tilde{B}$

Decompose $\tilde{v}$ into $\tilde{v}_{\bot}$ (parallel to $\tilde{B}$, no magnetic force) and $\tilde{v}_{\bot}$ (perpendicular to $\tilde{B}$, causes deflection).

Centripetal force role of magnetic force

When $\tilde{v} \bot \tilde{B}$, magnetic force acts as the centripetal force, changing direction of motion without changing speed.

Gyroradius (radius of circular motion)

For $v \bot B$: $r = \frac{mv}{|q|B}$; larger $m$ or $v$ increases $r$, larger $|q|$ or $B$ decreases $r$.

Cyclotron angular frequency ($\theta$)

Angular speed of circular motion in a uniform magnetic field: $\theta = \frac{|q|B}{m}$ (nonrelativistic).

Cyclotron period (T)

Time for one revolution: $T = \frac{2\theta m}{|q|B}$; independent of particle speed (nonrelativistic).

Helical motion

Motion when both $v_{\bot}$ and $v_{\bot}$ are present: circular motion from $v_{\bot}$ combined with constant translation along $\tilde{B}$ from $v_{\bot}$.

Velocity selector

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