Force on Moving Charge in a Magnetic Field | Class 12 Physics Derivation, Conditions, and Solutions

Force on Moving Charge in a Magnetic Field

Let us consider a positive electric charge 'q' that moves through a magnetic field (magnetic field strength B) with a velocity 'v' and makes an angle '$\theta$' with the magnetic field. Then, it experiences a force due to the field perpendicular to the velocity and the magnetic field.

Experimentally, we can observe that

1) Magnetic force is directly proportional to the magnitude of the charge

$\rm F \propto q$

2) Magnetic force is directly proportional to the velocity of the charge

$\rm F \propto v$

3) Magnetic force is directly proportional to the strength of the magnetic field

$\rm F \propto B$

4) Magnetic force is directly proportional to the sine of the angle between v and B

$\rm F \propto sin \theta$

From the above proportionalities, we get,

$\rm F \propto q v B \sin \theta$

$\rm or, F = k qv B \sin \theta$

where k is a proportionality constant and its value is 1 in SI units. So, we have force,

$\rm F = qvB \sin \theta$

In vector form,

$\rm F = q ( \vec{v} \times \vec{B} )$

Conditions of Force on a Moving Charge in a Magnetic Field

• If the velocity is parallel or antiparallel ($\theta = 0 or \theta = 180$ to the magnetic field, the field exerts a maximum magnetic force on the charge.
• If the velocity is perpendicular to the magnetic field, the field exerts no force on the charge.
• If $\rm v = 0$, then the particle experiences no force due to the magnetic field. This means only charges in motion experience a magnetic force.
• If $\rm q = 0$, then the particle experiences no force due to the magnetic field. This means electrically neutral particles do not observe a magnetic force.