A particle of charge q and mass m starts moving from the origin under the action of an electric field E→=Eoi^ and B→=Boi^ with a velocity ⇒ v→=v0 j^ . The speed of the particle will become 2vo after a time t=3 mvoWqE . Find W.

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A particle of charge $q$ and mass $m$ starts moving from the origin under the action of an electric field $\vec{E} = E_{o} \hat{i}$ and $\vec{B} = B_{o} \hat{i}$ with a velocity $\Rightarrow \vec{v} = v_{0} \hat{j}$ . The speed of the particle will become $2 v_{o}$ after a time $t = \frac{\sqrt{3} m v_{o}}{W q E}$ . Find $W$ .

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Detailed Solution

$\vec{E}$ is parallel to $\vec{B}$ and $\vec{v}$ is perpendicular to both. Therefore, path of the particle is a helix with increasing pitch. Speed of particle at any time $t$ is

$v = \sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}}$ (1)

here $v_{y}^{2} + v_{z}^{2} = v_{o}^{2}$ and $v = 2 v_{o}$

$v_{x} = \sqrt{3} v_{0}$

$t = \frac{\sqrt{3} v_{0}}{a} = \frac{\sqrt{3} v_{0} m}{q E}$

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