First slide

Force on a charged particle moving in a magnetic field

Question

A particle of mass m having negative charge $- q$ is projected at an angle $θ$ with x-axis. There exists a uniform electric field E and a uniform magnetic field B along x-axis. The particle will return to its initial point after n complete revolutions in time t. What is n here?

Difficult

A

$n = \frac{Bvcosθ}{πE}$
B

$n = \frac{2 Bvcosθ}{πE}$
C

$n = \frac{Bv \sin θ}{πE}$
D

$n = \frac{2 Bv \sin θ}{πE}$

Solution

Let particle completes n revolutions in time t, then t = nT
$\Rightarrow t = \frac{n 2 πm}{qB}$
If particle has to return to its initial point, then displacement along x-axis during this time should be zero.
$\begin{array}{l} S_{x} = u_{x} t + \frac{1}{2} a_{x} t^{2} \\ \Rightarrow 0 = v \cos θt - \frac{1}{2} \frac{qE}{m} t^{2} \\ \Rightarrow v \cos θ = \frac{qE}{2 m} t \\ \Rightarrow v \cos θ = \frac{qE}{2 m} \times \frac{n 2 πm}{qB} \Rightarrow n = \frac{Bvcosθ}{πE} \end{array}$

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