First slide

Force on a charged particle moving in a magnetic field

Question

The magnetic field in a region is given by $\vec{B} = B_{0} \hat{i} .$ A charged particle of specific charge $α$ is given a velocity $\vec{v} = (\frac{v_{0}}{2} \hat{i} + \frac{\sqrt{3} v_{0}}{2} \hat{j})$ at the origin. Find the minimum distance of the point on the x-axis from the origin where the particle will cut the x-axis.

Moderate

A

$\frac{\sqrt{3} π v_{0}}{2 α B_{0}}$
B

$\frac{π v_{0}}{2 α B_{0}}$
C

$\frac{π v_{0}}{α B_{0}}$
D

$\frac{\sqrt{3} π v_{0}}{α B_{0}}$

Solution

$\begin{array}{l} ω = \frac{Q B_{0}}{m} = α B_{0} ∴ T = \frac{2 π}{ω} = \frac{2 π}{α B_{0}} \\ ∴ M i n i m u m d i s t a n c e = T \times \frac{V_{0}}{2} = \frac{π V_{0}}{α B_{0}} \end{array}$

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