First slide

Motion of a charged particle

Question

A particle of mass m and q has an initial velocity $\vec{v} = v_{0} \hat{j}$ . If an electric field $\vec{E} = E_{0} \hat{i}$ and magnetic field $\vec{B} = B_{0} \hat{i}$ act on the particle, its speed will double after a time:

Difficult

A

$\frac{\sqrt{3} m v_{0}}{q E_{0}}$
B

$\frac{2 m v_{0}}{q E_{0}}$
C

$\frac{3 m v_{0}}{q E_{0}}$
D

$\frac{\sqrt{2} m v_{0}}{q E_{0}}$

Solution

$\begin{array}{l} S i n c e, \vec{B} i s p e r p e n d i c u l a r t o i n i t i a l v e l o c i t y, m a g n i t u d e o f v e l o c i t y d o e s n o t c h a n g e i n y - z p l a n e . \\ \Rightarrow {v_{o}}^{2} = {v_{y}}^{2} + {v_{z}}^{2} \\ a n d \vec{E} c a n a c c e l e r a t e p a r t i c l e o n l y a l o n g x a x i s . \\ \Rightarrow a_{x} = \frac{q E_{0}}{m} \\ A t a n y g i v e n t i m e s p e e d o f p a r t i c l e i s, v = \sqrt{{v_{x}}^{2} + {v_{y}}^{2} + {v_{z}}^{2}} \\ W h e n s p e e d b e c o m e s d o u b l e, w e c a n w r i t e \\ 2 v_{o} = \sqrt{{v_{x}}^{2} + {v_{y}}^{2} + {v_{z}}^{2}} \\ \Rightarrow 2 v_{o} = \sqrt{{v_{x}}^{2} + {v_{o}}^{2}} \\ \Rightarrow v_{x} = \sqrt{3} v_{o} \end{array}$
$\begin{array}{l} U \sin g e q u a t i o n o f k i n e m a t i c s, \\ v_{x} = u_{x} + a_{x} t \\ \Rightarrow \sqrt{3} v_{0} = 0 + \frac{q E_{0}}{m} t \\ \Rightarrow t = \frac{\sqrt{3} m v_{0}}{q E_{0}} \end{array}$

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