Lorentz Force: Force on a Moving Charge in Electric and Magnetic Fields

Force due to a magnetic field alone. A charge $q$ moving with velocity $\vec{v}$ through a

region containing a magnetic field $\vec{B}$ experiences a magnetic force

with magnitude $F_m = qvB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.

Because this is a vector CROSS product, the magnetic force is always perpendicular to BOTH the

particle's velocity and the field -- and, since it always acts at right angles to the velocity, it

can never have any component along the direction of motion. A force with no component along the

motion does zero work ($dW = \vec{F}\cdot d\vec{s} = 0$ at every instant, since $\vec{F}_m \perp \vec{v}$ always): the magnetic force can change a moving charge's DIRECTION, but never its SPEED or kinetic energy. Two special cases follow immediately from the $\sin\theta$ factor: if

$\vec{v}$ is exactly PARALLEL (or antiparallel) to $\vec{B}$, then $\theta=0$ or $180^\circ$ and

$F_m=0$ -- a charge moving exactly along the field lines feels no magnetic force at all; if

$\vec{v}$ is exactly PERPENDICULAR to $\vec{B}$ ($\theta=90^\circ$), the force is at its maximum,

$F_m = qvB$, and, being always perpendicular to $\vec{v}$, this maximal case produces UNIFORM

CIRCULAR MOTION -- the situation examined in full in Section 4.9.

The full Lorentz force: adding an electric field. If an electric field $\vec{E}$ is ALSO present

in the same region, the charge experiences the ordinary electrostatic force $q\vec{E}$ in addition

to the magnetic force above. The TOTAL electromagnetic force, called the Lorentz force, is the

vector sum of the two:

Unlike the magnetic force alone, the electric-field contribution $q\vec{E}$ is entirely independent

of the charge's velocity and CAN do work on the charge, changing its kinetic energy -- an electric

field is what actually accelerates or decelerates a charged particle, while a magnetic field can only

steer it.

A direct application: the velocity selector. A particularly useful configuration has $\vec{E}$

and $\vec{B}$ set up mutually perpendicular to each other, and both perpendicular to the charge's

initial velocity $\vec{v}$, with their magnitudes and directions arranged so the electric force

$q E$ and the magnetic force $qvB$ act in exactly OPPOSITE directions. For most charges entering

such a region, these two forces do not balance, and the charge is deflected off its original straight …