Question

A charged particle is released from origin with initial velocity $V=V_0\hat{i}$ in a magnetic field $\overrightarrow{B}=\frac{\sqrt{3}}{2}{B}_0\hat{i}+\frac{B_0}{2}\hat{j}$. The charge and mass of the particle are $q$ and $m$ respectively. Choose the correct option(s):

• A. the pitch of the helical path described by the particle is $\frac{\pi m{V}_0\sqrt{3}}{B_{0}q}$
• B. the pitch of the helical path described by the particle is $\frac{2\pi m{V}_0}{B_{0}q}$
• C. the velocity $\frac{V_0\sqrt{3}}{2}$ is responsible for progress of particle in direction of magnetic field
• D. velocity $\frac{V_0}{2}$ is responsible for circular motion of particle in helix

Answer 1

Answer: (A), (C), (D)

the velocity $\frac{V_0\sqrt{3}}{2}$ is responsible for progress of particle in direction of magnetic field
velocity $\frac{V_0}{2}$ is responsible for circular motion of particle in helix
the pitch of the helical path described by the particle is $\frac{\pi m{V}_0\sqrt{3}}{B_{0}q}$

Time period of the charged particle in circular motion: $T=\frac{2\pi m}{q{B}_0}$
Component of velocity along the direction of magnetic field:
$\frac{\overrightarrow{V}.\overrightarrow{B}}{\left|\overrightarrow{B}\right|}=\left|\overrightarrow{V}\right|\frac{\sqrt{3}}{2}=\frac{\sqrt{3}{V}_0}{2}$

Pitch is the distance covered in time period T along the direction of magnetic field.
$\therefore$ Pitch, $P=\frac{\sqrt{3}V}{2}\times T=\frac{\sqrt{3}{V}_0}{2}\times \frac{2\pi m}{q{B}_0}$
Component of velocity perpendicular to magnetic field is responsible for motion in a circular path
$V_{\perp }=\frac{V_0}{2}$