First slide

Force on a charged particle moving in a magnetic field

Question

A particle of mass m and charge q moves with a constant velocity v along the positive x-direction. It enters a region containing a uniform magnetic field B directed along the negative z-direction, extending from x = a to x = b. The minimum value of v required so that the particle can just enter the region x > b, is

Moderate

A

qbB/m
B

$q (b - a) B / m$
C

qaB/m
D

q(b + a)B/2m

Solution

In the figure, the z-axis points out of the paper, and the magnetic field is directed into the paper, existing in the region between PQ and RS. The particle moves in a circular path of radius r in the magnetic field. It can just enter the region $x > b for r \geq (b - a)$
$\begin{array}{l} Now, r = \frac{mv}{qB} \geq (b - a) \\ or v \geq \frac{q (b - a) B}{m} \Rightarrow v_{\min} = \frac{q (b - a) B}{m} \end{array}$

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