 # Force on Electric Charges Moving in Electric and Magnetic Fields

### What is the Force due to a Magnetic Field?

Magnetic fields can exert a force on electric charge only if it is moves, just as a moving charge produces a magnetic field. This force increases with both an increase in charge as well as magnetic field strength. Moreover, the force is greater when charges have higher velocities.

The magnetic force however, always acts perpendicular to the velocity. Thus this force can never produce work on the charge and cannot impart it any kinetic energy. The magnetic force is given by:

$\overrightarrow{F_m}$ = $q~\overrightarrow{v}~×~\overrightarrow{B}$

Where $q$ is the charge, $v$ is the velocity and $B$ is the magnetic field. Notice that the cross product implies that the force always acts perpendicular to both the velocity and magnetic field. Thus, it always acts out of plane and does not contribute to any work done on the charge. It can merely change the direction of the velocity but cannot change its magnitude. The direction of force can be easily determined using Fleming’s Right Hand Rule.

### What is the Force Due to Electric Field?

The force due to electric field on a charge is on built into its definition. It always acts either parallel or anti parallel to the electric field and is independent of the velocity of charge. This means it has the ability to do work and impart energy to the charge.

$\overrightarrow{F_e}$  =  $q ~\overrightarrow{E}$

### What is Lorentz Force?

When a charge travels through both an electric and magnetic field, the net force on the charge is called the Lorentz force. It is simply the sum of the magnetic and electric forces:

$\overrightarrow{F}$ = $\overrightarrow{F_e}~+~\overrightarrow{F_m}$

$\overrightarrow{F}$ = $q~\overrightarrow{E}~+~q~\overrightarrow{v}~×~\overrightarrow{B}$

$\overrightarrow{F}$ = $q(\overrightarrow{E}~+~\overrightarrow{v}~×~\overrightarrow{B}$)

Combinations of electric and magnetic fields are used in particle accelerators, cyclotrons and synchrotrons. The magnetic field can be used to keep the charges moving in a circle while the electric field is used to accelerate the charges and impart them energy.