Magnetic Force on Charges & Currents Cheat Sheet

Key Facts

• The magnetic force on a moving charge has magnitude $F_B = |q|vB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.
• The vector form of magnetic force on a charge is $\vec{F}_B = q\vec{v} \times \vec{B}$, so a negative charge feels force opposite the right-hand-rule direction.
• A charge moving parallel or antiparallel to a magnetic field feels no magnetic force because $\sin 0^{\circ} = 0$ and $\sin 180^{\circ} = 0$.
• A charge moving perpendicular to a uniform magnetic field follows circular motion when magnetic force supplies centripetal force, so $|q|vB = \frac{mv^2}{r}$.
• The radius of circular motion in a uniform magnetic field is $r = \frac{mv}{|q|B}$.
• The angular speed and period for perpendicular circular motion are $\omega = \frac{|q|B}{m}$ and $T = \frac{2\pi m}{|q|B}$.
• The magnetic force on a straight current-carrying wire is $F_B = ILB\sin\theta$, where $L$ is the wire length inside the magnetic field.
• The torque on a current loop is $\tau = NIAB\sin\theta$, where $N$ is the number of turns and $A$ is the loop area.

Vocabulary

Magnetic field

A region described by $\vec{B}$ where moving charges or currents can experience magnetic force.

Lorentz force

The total electromagnetic force on a charge, often written $\vec{F} = q\vec{E} + q\vec{v} \times \vec{B}$.

Right-hand rule

A direction rule where fingers follow $\vec{v}$ or current, curl toward $\vec{B}$, and the thumb gives the force direction for positive charge or conventional current.

Centripetal force

The inward net force needed for circular motion, with magnitude $F_c = \frac{mv^2}{r}$.

Conventional current

The direction of current defined as the motion of positive charge, even though electrons in metals move the opposite way.

Magnetic torque

The turning effect on a current loop in a magnetic field, with magnitude $\tau = NIAB\sin\theta$.