Question

A Charged particle of mass \[m\] and charge \[q\] is accelerated through a potential difference of \[V\]\[volt\]. It enters a region of uniform magnetic field which is directed perpendicular to the direction of motion of the particle. The particle will move on a circular path of radius given by :
\[
  {\text{(A) }}\dfrac{{Vm}}{{q{B^2}}} \\
  {\text{(B) }}\dfrac{{2Vm}}{{q{B^2}}} \\
  {\text{(C) }}\sqrt {\dfrac{{2Vm}}{q}} \left( {\dfrac{1}{B}} \right) \\
  {\text{(D) }}\sqrt {\dfrac{{Vm}}{q}} \left( {\dfrac{1}{B}} \right) \\
\]

Answer 1

Hint: Calculate the final velocity of the body after it accelerates by a potential difference of \[V\]\[volt\]. Then equate the force due to the magnetic field to the centripetal force required to move in a circle.

Complete step by step answer:
The work done \[(W)\] when charge \[q\] is accelerated through a potential difference of \[V\]\[volt\]is :
$W = qV$ …. (A)
This work done is stored in the form of Kinetic energy. Now let v be the final velocity of the body as it enters the magnetic field. We have :
$
{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}m{v^2} = qV \\
v = \sqrt {\dfrac{{2qV}}{m}} \\
 $
Since the direction of the magnetic field is perpendicular to the direction of motion of the particle, it starts moving in a circular path. The necessary centripetal force required to move in the circle is provided by the force exerted by the magnetic field on the charged particle.
Let r be the radius of the path followed by the body. Then we have :
$
\Rightarrow \dfrac{{m{v^2}}}{r} = qvB \\
\Rightarrow \dfrac{{mv}}{r} = qB \\
\Rightarrow r = \dfrac{{mv}}{{qB}} \\
\therefore r = \dfrac{m}{{qB}}\sqrt {\dfrac{{2qV}}{m}} = \sqrt {\dfrac{{2mV}}{q}} \left( {\dfrac{1}{B}} \right) \\
 $

So, the correct answer is “Option C”.

Additional Information:
Since the force exerted by the magnetic field is always perpendicular to the direction of velocity of the particle, the magnetic force cannot change the magnitude of velocity. It can only change the direction of velocity.

Note:
Whenever a body enters a region of magnetic field with its velocity perpendicular to the field, the path followed by the body is circular.
The direction of force exerted by the magnetic field is given by right hand screw rule.