Electric field lines between charged plates and magnetic field patterns

Magnetic force on charges and currents

PHYS 301 · Magnetostatics

Magnetic fields exert forces on moving charges and current-carrying wires. This lesson introduces the Lorentz force, circular motion, and force on currents.

Key equations

\vec{F}_B=q\vec{v}\times\vec{B}F_B=|q|vB\sin\thetaP=\vec{F}\cdot\vec{v}=0|q|vB=m\frac{v^2}{r}r=\frac{mv}{|q|B}\omega_c=\frac{|q|B}{m}\vec{F}=q(\vec{E}+\vec{v}\times\vec{B})\vec{F}=I\vec{L}\times\vec{B}F=ILB\vec{\mu}=IA\hat{n}\vec{\tau}=\vec{\mu}\times\vec{B}

Learning objectives

Apply the magnetic force law for moving charges.
Use right-hand rules for magnetic force directions.
Analyze circular and helical motion in magnetic fields.
Apply the Lorentz force law.
Calculate forces and torques on current-carrying conductors.

Magnetic fields and moving charges

A magnetic field, written $ec{B}$ , exerts force on moving electric charges. Unlike an electric field, a magnetic field does not exert force on a stationary charge. The magnetic force on a charge $q$ moving with velocity $ec{v}$ is

$ec{F}_B=q ec{v} imes ec{B}$

The magnitude is

$F_B=|q|vBsin heta$

where $heta$ is the angle between velocity and magnetic field.

The direction is given by the right-hand rule for a positive charge. For a negative charge, the force is opposite the right-hand-rule direction.

Magnetic force is perpendicular

The magnetic force is always perpendicular to both $ec{v}$ and $ec{B}$ . Therefore, it does no work on a charged particle:

$P= ec{F}cdot ec{v}=0$

A magnetic field can change the direction of a particle's velocity, but it cannot change its speed or kinetic energy by itself.

Circular motion in a uniform magnetic field

If a charged particle enters a uniform magnetic field with velocity perpendicular to the field, the magnetic force provides centripetal force:

$|q|vB=m rac{v^2}{r}$

Solving for radius,

$r= rac{mv}{|q|B}$

The angular frequency of circular motion is

$omega_c= rac{|q|B}{m}$

This is called the cyclotron frequency.

Helical motion

If the particle's velocity has a component parallel to the magnetic field, that component is unaffected by the magnetic force. The perpendicular component produces circular motion, while the parallel component carries the particle along the field. The result is a helix.

This motion occurs in particle accelerators, mass spectrometers, plasmas, and charged particles trapped by Earth's magnetic field.

Lorentz force

When both electric and magnetic fields are present, the total electromagnetic force is

$ec{F}=q( ec{E}+ ec{v} imes ec{B})$

This is the Lorentz force law. It is one of the central equations of electromagnetism.

Force on a current-carrying wire

A current is moving charge. A wire segment of length vector $ec{L}$ carrying current $I$ in a magnetic field experiences force

$ec{F}=I ec{L} imes ec{B}$

For a straight wire perpendicular to a uniform field,

$F=ILB$

If the wire is parallel to the field, the force is zero.

Torque on a current loop

A current loop in a magnetic field experiences torque. Its magnetic dipole moment is

$ec{mu}=IAhat{n}$

where $A$ is loop area and $hat{n}$ is the area direction from the right-hand rule. The torque is

$ec{ au}= ec{mu} imes ec{B}$

This principle underlies electric motors and analog meters.

The big idea

Magnetic fields act on moving charges and currents through cross products. The force is perpendicular to motion and therefore changes direction rather than speed. Magnetic forces produce circular and helical particle motion, forces on wires, and torques on current loops, making them essential in motors, accelerators, and magnetic devices.

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