First slide

Force on a charged particle moving in a magnetic field

Question

A proton is projected from the origin making an angle of 60° with the x-axis. A magnetic field $\vec{B} = 2 \hat{i}$ tesla exists in that region of space. The proton passes through the x-axis for the first time at a distance of 50 cm from the origin. When an α - particle is projected from the origin with same speed making an angle of 30° with the x-axis in the same magnetic field, it passes though the x-axis at a distance ‘d’ from the origin. Then the value of ‘d’ is

Difficult

A

$10 \sqrt{3} cm$
B

$100 \sqrt{3} cm$
C

$200 \sqrt{3} cm$
D

$20 \sqrt{3} cm$

Solution

Trajectory of the proton or α - particle is helix whose pitch is given by $p = \frac{2 πmVcosα}{QB}$
$\begin{array}{rcl} ∴ \frac{p_{α}}{p_{p}} & = & (\frac{m_{α}}{m_{p}}) (\frac{Q_{p}}{Q_{α}}) \frac{{(\cos α)}_{α}}{{(\cos α)}_{p}} \Rightarrow \frac{d}{50} = (4) (\frac{1}{2}) (\frac{\cos 30^{0}}{\cos 60^{o}}) \\ \Rightarrow & d = 100 \sqrt{3} cm \end{array}$

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