Question

A particle of mass m and positive charge q, moving with a constant velocity ${{u}_{1}}=4\hat{i}m{{s}^{-1}}$, enters a region of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from x=0 to x=l for all values of y. After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity \[{{u}_{2}}=2(\sqrt{3}\hat{i}+\hat{j})m{{s}^{-1}}\]. The correct statement(s) is (are)
A) the direction of magnetic field is -z direction
B) the direction of magnetic field is +z direction
C) the magnitude of magnetic field $\dfrac{50\pi M}{3Q}$ units
D) the magnitude of the magnetic field $\dfrac{100\pi M}{3Q}$ units.

Answer 1

Hint: Let us find the time spent by the particle inside the magnetic field. Write down the relation between t and T. Next, time taken by the particle to come out of the magnetic field is given, we can easily find the magnetic field formula in terms of known quantities.

Formulas used:
$T=\dfrac{2\pi M}{qB}$

Complete answer:
Time period of the circular motion is given as $T=\dfrac{2\pi M}{qB}$
Time spent by the particle inside the magnetic field is given as,
$\begin{align}
  & t=\dfrac{\pi }{6}\times T \\
 & \Rightarrow t=\dfrac{\pi M}{qB}....(1) \\
 & \Rightarrow t=10ms=10\times {{10}^{-3}}s....(2) \\
\end{align}$
As equation one is equal to equation two,
$\begin{align}
  & 10\times {{10}^{-3}}s=\dfrac{\pi M}{qB} \\
 & B=\dfrac{50\pi M}{QB} \\
\end{align}$
As seen in the given picture, velocity is $4\hat{i}m{{s}^{-1}}$. Since the particle bends upward, the magnetic field is in the direction of the negative z axis.

Additional information:
A charged particle experiences a force when moving through a magnetic field. The magnetic force acting on the particle is perpendicular to the direction of travel; a charged particle follows a curved path in a magnetic field. If the field is in a vacuum, the magnetic field is the dominant factor determining the motion. The particle continues to follow this code path until it forms a complete circle. We can easily say that the magnetic force is always perpendicular to the velocity, so that it does not work on charged particles. The particle's kinetic energy and the speed remains constant in the magnetic field. Direction of motion is affected but not the speed of the particle.

Note:
When a particle is present in a magnetic field, the magnetic field does not affect the speed and it cannot affect the speed of the particle. This is because the magnetic field applies a force perpendicular to the velocity always. The force can't do work on the particle, and there will not be any change in the kinetic energy meaning it cannot change the speed.