First slide

Force on a charged particle moving in a magnetic field

Question

There exists a uniform magnetic and electric field of magnitude 1 T and 1 V /m, respectively, along positive y-axis. A charged particle of mass 1 kg and of charge 1 C is having velocity 1 m/s along x-axis and is at origin at t = 0. Then the co-ordinates of particle at $time - π$ seconds will be

Difficult

A

(0, 1, 2) m
B

$(0, π^{2}, - 2) m$
C

$(2, π^{2} / 2, 2) m$
D

$(0, π^{2} / 2, 2) m$

Solution

The particle will move in a non-uniform helical path with increasing pitch as shown below.
Its time period will be: $T = \frac{2 πm}{qB} = 2 π seconds$
Changing the view, the particle is seemed to move in a circular path in (X - Z) plane as shown below
After $π$ -seconds the particle will be at point 'P' (after completing half circle), hence X coordinate of P will be 0.
$\begin{array}{l} y (π) = 0 (π) + \frac{1}{2} \frac{Eq}{m} {(π)}^{2} = \frac{π^{2}}{2} \\ z = 2 r = 2 mv / qB = 2 m \\ Hence, the coordinate (0, \frac{π^{2}}{2}, 2) \end{array}$

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