First slide

Motion of a charged particle

Question

A charged particle of specific charge $(\frac{q}{m}) = α$ is released from origin at time $t = 0$ with velocity $\bar{V} = V_{0} (\hat{i} + \hat{j})$ in uniform magnetic field $\bar{B} = B_{0} \hat{i}$ . Co-ordinates of the particle at time $t = \frac{π}{B_{0} α}$ are.

Difficult

A

$(\frac{v_{0}}{2 B_{0} α}, \frac{\sqrt{2} . v_{0}}{α B_{0}}, \frac{- v_{0}}{B_{0} α})$
B

$(\frac{- v_{0}}{2 B_{0} α}, 0, 0)$
C

$(0, \frac{2 v_{0}}{B_{0} α}, \frac{v_{0} π}{2 B_{0} α})$
D

$(\frac{v_{0} π}{B_{0} α}, 0, \frac{- 2 v_{0}}{B_{0} α})$

Solution

$α = \frac{q}{m}$ ,
Path of the particle will be a helix of time period
$T = \frac{2 π m}{B_{o} q} = \frac{2 π}{B_{0} α}$

The given time $t = \frac{π}{B_{0} α} = \frac{T}{2}$
$∴$ Coordinates of particle at time $t = \frac{T}{2}$
Would be $(V_{x} T / 2, 0, - 2 r)$
Here, $r = \frac{m V_{0}}{B_{0} q} = \frac{v_{0}}{B_{0} α}$
$∴$ The coordinates are $(\frac{V_{o} π}{B_{0} α}, 0, \frac{- 2 V_{0}}{B_{o} α})$ .

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