Question

A particle of specific charge \( \frac{q}{m} \) is projected from the origin towards the positive x-axis with the velocity \( 10 \, \text{m/s} \) in a uniform magnetic field \( B = -2 \hat{k} \, \text{T} \). The velocity \( v \) of the particle after time \( t = 5 \, \text{ms} \) will be (in m/s)

• A. \( 5\hat{i} + 5\hat{j} \)
• B. \( 5\hat{i} - 5\hat{j} \)
• C. \( 5\hat{i} + 5\hat{k} \)
• D. \( 5\hat{i} - 5\hat{k} \)

Answer 1

The Correct Option is A

The time period $T$ is given by:
\[ T = \frac{2\pi m}{qB} = \frac{2\pi}{m \times 2} = 15 \] 

The particle will be at point $P$ after time: 
\[ t = \frac{1}{12} \, \text{s} = \frac{T}{12} \] 

It is deviated by an angle: 
\[ \theta = \frac{2\pi}{12} = 30^\circ \] 

Thus, the velocity of the particle at point $P$ is: 
\[ \vec{v} = 10\cos 30^\circ \hat{i} + 10\sin 30^\circ \hat{j} \] \[ \vec{v} = 10 \left(\frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j} \right) \] \[ \vec{v} = 5\left(\sqrt{3} \hat{i} + \hat{j}\right) \]