Motion of a Charged Particle in a Magnetic Field
The motion of a charged particle when it is moving collinear with the field, the magnetic field is not affected by the field (i.e. if motion is just along or opposite to a magnetic field) ( $ \quad F=0$ ) Only the following two cases are possible:
The case I: When the charged particle is moving perpendicular to the field The angle between $\overrightarrow{\mathrm{B}}$ and $\overrightarrow{\mathrm{v}}$ is $\theta=90^{\circ}$
So the force will be maximum ( $=$ qvB ) and always perpendicular to motion (and also field);
Hence the charged particle will move along a circular path (with its plane perpendicular to the field).
Centripetal force is provided by the force qvB,


In case of the circular motion of a charged particle in a steady magnetic field :

i.e., with the increase in speed or kinetic energy, the radius of the orbit increases. For uniform circular motion $v=\omega r$
Angular frequency of circular motion called cyclotron or gyro-frequency. $\omega=\frac{\mathrm{v}}{\mathrm{r}}=\frac{\mathrm{qB}}{\mathrm{m}}$
and the time period, $\quad \mathrm{T}=\frac{2 \pi}{\omega}=2 \pi \frac{\mathrm{m}}{\mathrm{qB}}$
i.e., time period (or frequency) is independent of speed of particle and radius of the orbit.
Time period depends only on the field B and the nature of the particle,
i.e., specific charge (q/m) of the particle.
This principle has been used in a large number of devices such as cyclotron (a particle accelerator), bubble-chamber (a particle detector) or mass-spectrometer etc.
the motion of a charged particle in an electric and magnetic field
case II : The charged particle is moving at an angle $\theta$ to the field :
$\left(\theta \neq 0^{\circ}, 90^{\circ} \text { or } 180^{\circ}\right)$
Resolving the velocity of the particle along and perpendicular to the field.
The particle moves with constant velocity v cos $\theta$ along the field ($\because$ no force acts on a charged particle when it moves parallel to the field).
And at the same time, it is also moving with velocity $v$ sin $\theta$ perpendicular to the field due to which it will describe a circle (in a plane perpendicular to the field)



So the resultant path will be a helix with its axis parallel to the field $\overrightarrow{\mathrm{B}}$ as shown in fig. The pitch p of the helix $=$ linear distance travelled in one rotation
$p=T(v \cos \theta)=\frac{2 \pi m}{q B}(v \cos \theta)$
Also Read:
Biot Savart’s Law
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