Answer to Question #134982 in Electricity and Magnetism for Max

Question

Explain the lorentz force.

Answer 1

The Lorentz force is the force with which an electromagnetic field acts on a point charged particle according to classical (non — quantum) electrodynamics.

The force F acting on a particle with an electric charge q moving at a speed v in the external electric E and magnetic B fields is:

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

The electromagnetic force on a charge q is a combination of a force in the direction of the electric field E proportional to the magnitude of the field and the quantity of charge, and a force at right angles to the magnetic field B and the velocity v of the charge, proportional to the magnitude of the field, the charge, and the velocity. (see https://en.wikipedia.org/wiki/Lorentz_force)

More explicitly:

$\vec{F}(\vec{r},t,q) = q\vec{E}(\vec{r},t) + q\dot{\vec{r}}\times \vec{B}(\vec{r},t),$

where r is the radius vector of the charged particle, t is the time, and the point denotes the derivative with respect to time.

For a continuous charge distribution, the Lorentz force takes the form

$d\vec{F} = dq(\vec{E}+\vec{v}\times\vec{B})$,

where dF is the force acting on the small element dq

"In many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an electron or ion in a plasma) can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation." (see https://en.wikipedia.org/wiki/Lorentz_force)