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

(a) $\displaystyle 5[\hat{i} + \sqrt 3 \hat{j}] $

(b) $ \displaystyle 5[\sqrt 3 \hat{i} + \hat{j}] $

(c) $ \displaystyle 5[\sqrt 3 \hat{i} – \hat{j}] $

(d) $\displaystyle 5[\hat{i} + \hat{j}] $

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$ \displaystyle T = \frac{2\pi m}{q B} $

$ \displaystyle T = \frac{2\pi}{(q/m) B} $

$ \displaystyle T = \frac{2\pi }{\pi \times 2} = 1 s $

after 1/12 sec , θ = 360°/12 = 30°

$ \displaystyle \vec{v} = 10 cos30\hat{i}+ 10sin30\hat{j} $

$ \displaystyle \vec{v} = 10 (cos30\hat{i}+ sin30\hat{j}) $

$ \displaystyle \vec{v} = 10 (\frac{\sqrt{3}}{2}\hat{i}+ \frac{1}{2}\hat{j}) $

$ \displaystyle \vec{v} = 5 (\sqrt{3}\hat{i}+ \hat{j}) $