Question

A particle of specific charge (q/m) is projected from the origin of coordinates with initial velocity [ui – vj]. uniform electric magnetic fields exist in the region along the +y direction, of magnitude e and

b.the particle will definitely return to the origin once if

Answer 1

In Y direction the electric field E exist

so the force due to electric field on the charge is given by

$F = qE$

Now the acceleration of the charge is given by

$a = \frac{qE}{m}$

now the acceleration is opposite to the direction of velocity

so it will return to origin if displacement in y direction is zero

$d = v_y * t + \frac{1}{2} at^2$

$0 = -v * T +\frac{1}{2}\frac{qE}{m}T^2$

$T = \frac{2mv}{qE}$

now due to magnetic field it will move in circular path

time period of its circular motion is given by

$T = \frac{2\pi m}{qB}$

now if complete N number of rounds in above time of return then

$N *\frac{2\pi m}{qB} = \frac{2mv}{qE}$

$N = \frac{Bv}{\pi E}$

so here above value must be an integer so that it complete integral nuymber of rounds

$integer = \frac{Bv}{\pi E}$

so above condition is must to return at origin

Answer 2

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