Motion of a Charged Particle in a Uniform Magnetic Field —
In uniform magnetic field, force experienced by a moving charged particle is given by
$\vec{F}=q(\vec{v} \times \vec{B}) \text { or } F=q v B \sin \theta$
Here, $\vec{v}$ is the velocity of the particle and $\vec{B}$ is uniform magnetic field and $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. The following conclusions are drawn from this expression:
- Stationary charge (i.e. $\vec{v}=0$ ) experiences no magnetic force.
- If $\vec{v}$ is parallel or antiparallel to $\vec{B}$ then the charged particle experiences no magnetic force.
- As magnetic force is always perpendicular to $\vec{v}$, it does not deliver any power to the charged particle.
- Magnetic force is always perpendicular to both $\vec{v}$ and $\vec{B}$.
- The maximum value of magnetic force is equal to $q v B$, which occurs when the charge particle is projected perpendicular to the uniform magnetic field.
➤ In this case, path of charged particle is circular and magnetic force provides the necessary centripetal force.
➤If radius of the circular path is $R$, then $\frac{m v^2}{R}=q v B$, where $m$ is the mass of the particle.
$\therefore \quad R=\frac{m v}{q B}$
➤ Time taken to complete one revolution is
$T=\frac{2 \pi R}{v} \text { or } T=\frac{2 \pi m}{q B}$
- If the charged particle is projected obliquely to the field then its velocity $\vec{v}$ can be resolved into two components; one along $\vec{B}$ say $v_{\|}$and the other perpendicular to $\vec{B}$ say $v_{\perp}$. It experiences a magnetic force and hence has a tendency to move on a circular path. Due to $v_{\|}$it experiences no force, and hence has a tendency to move on a straight path along the field. So, in this case it moves along a helical path.
➤Radius of the helix is $R=\frac{m v_{\perp}}{q B}$.
➤Time taken to complete one revolution is $T=\frac{2 \pi m}{a B}$.
➤The distances moved by the charged particle along the magnetic field during one revolution is called pitch.
$\text { Pitch }=v_{\|} \times T =\frac{2 \pi}{q B} m v_{\|}$
