Question

A particle of specific charge $\frac{q}{m}=\pi Ck{g}^{-1}$ is projected from the origin toward positive $x-$axis with a velocity of $10m{s}^{-1}$ in a uniform magnetic field $\overrightarrow{B}=-2\hat{k}T$. The velocity $\overrightarrow{v}$ of particle after time $t=\frac{1}{12}s$ will be $($in $m{s}^{-1})$

• A. $5[\hat{i}+\sqrt{3}\hat{j}]$
• B. $5[\sqrt{3}\hat{i}+\hat{j}]$
• C. $5[\sqrt{3}\hat{i}-\hat{j}]$
• D. $5[\hat{i}+\hat{j}]$

Answer 1

Answer: (B)

$5[\sqrt{3}\hat{i}+\hat{j}]$

Time period $T=\frac{2\pi m}{qB}=\frac{2\pi }{\frac{q}{m}B}=\frac{2\pi }{\pi \times 2}=1s$

Since the particle will be at point $P$ after time $t=\frac{1}{12}=\frac{T}{12}s$ , it is deviated by an angle $\theta =\frac{2\pi }{12}={30}^0$

Hence, velocity at point $P$

$\overrightarrow{v}=10(\cos {30}^0\hat{i}+\sin {30}^0\hat{j})=10(\frac{\sqrt{3}}{2}\hat{i}+\frac{1}{2}\hat{j})=5(\sqrt{3}\hat{i}+1\hat{j})=5(\sqrt{3}\hat{i}+\hat{j})$