Uniform electric field E and magnetic field B, respectively, are along y-axis as shown in the figure. A particle with specific charge q/m leaves the origin O in the direction of x-axis with an initial non-relativistic speed `v_(0). The coordinate `Y_(n)` of the particle when it crosses the y-axis for the nth time is:

Options

A. `(2pi^(2)mn^(2)E)/(qB^(2))`
B. `(pi^(2)mn^(2)E)/(qB^(2))`
C. `(2pi^(2)mn^(2)E)/(3qB^(2))`
D. `(sqrt(3)pi^(2)mn^(2) E)/(qB^(2))`

Answer

A

Explanation

`(dv_(y))/(dt)=(qE)/m=` constant
constant
The motion of the particle is equivalent to circular motion in xz plane with uniform acceleration in
y-direction. The magnetic force cannot change the magnitude of velocity. The y component of velocity,
`v_(y)=(qE)/mt,` The y-coordinate at time `y=1/2 (qE)/mt^(2)`
The time period of circular motion in xz plane `T=(2pim)/(Bq)`
Let the particle cross y-axis after n rotations, then `T=(2pim)/(Bq)`
Thus, `y_(n)=(qE)/(2m)xx((2pimn)/(qB))^(2)=(2pi^(2)mn^(2)E)/(qB^(2))`