Force on electric charge moving in a magnetic field 

In this lesson, we will learn:

• Magnitude and direction of an electromagnetic force exerted on moving charge in a magnetic field.
• The radius of the circular motion of a moving charged particle.

• A current-carrying wire experience a force when is placed in a magnetic field, similarly, freely moving charged particles would also experience a force when passing through a magnetic field.
• The magnitude of the electromagnetic force exerted on moving charge in a magnetic field is calculated using the following equation:
  
  $F = qvB \sin \theta$

• The force is greatest when the angle between $\overrightarrow{B}$ and $\overrightarrow{v}$ is 90°, $\qquad F_{max} = qvB$
• The force is zero if the particle moves parallel to the field lines and the angle between $\overrightarrow{B}$ and $\overrightarrow{v}$ is 0°


• The direction of the force is perpendicular to the magnetic field $\overrightarrow{B}$ and to the velocity $\overrightarrow{v}$ of the particle. 
• The direction of the force is found using the right-hand rule, fingers point along the direction of the particle’s velocity and bend your fingers towards the of $\overrightarrow{B}$. Then your thumb will point in the direction of the force. This is true only for positively charged particles, for negatively charged particles, the force is in exactly the opposite direction.

• The force exerted by a uniform magnetic field on a moving charged particle, produces a circular path.

• The particle would move in a circular path with constant centripetal acceleration if the force is always perpendicular to its velocity, $a = \frac{v^{2}}{r}$
• The force is greatest when the angle between $\overrightarrow{B}$ and $\overrightarrow{v}$ is 90°, $\qquad F_{max} = qvB$


$\sum F = ma$
$qvB = m \frac{v^{2}}{r}$
$r = \frac{mv} {qB}$
Since $\overrightarrow{F}$ is perpendicular to $\overrightarrow{v}$, the magnitude of $\overrightarrow{v}$ does not change. From this equation, we see that if $\overrightarrow{B}$ = constant, then $r$ = constant, and the curve must be a circle.