A charged point particle having positive charge `q`, mass `m` is projected in `x-z` plane with a speed `v_(0)` at an angle `theta` with the `x-"axis"` , while another identical particle is projected simultaneously along the `y-"axis"` from the origin (see figure) with an equal speed `v_(0)`.
An electric field `vecE= -E_(0)hatk` exists in the `x-z` plane, while a magnetic field `vecB=B_(0)hatk` exists in the region `y gt 0` (`F_(0)`, `B_(0) gt 0`). It is observed that both the particles collide after some time. Assume that the space is gravity free.
(`i`) Find the time after which the particles collide and the coordinates of the point of collision.
(`ii`) Find the velocity `v_(0)` and the angle `theta` in terms of the other variables.
(`iii`) What is the maximum value of the `z-` coordinate of the first particle ?

Explanation

The acceleration on the charge `q`, projected along the `x-z` palne is along the `-z` direction.
`a=(qE_(0))/(m)`
The second particle moves on the `x-y` plane in a circle of radius `R` (say) such that :
`2R=(2v_(0)^(2))/(a)sintheta costheta`.......(`i`)
Also, `R=(mv_(0))/(qB_(0))`.....(`ii`)
`T=(2v_(0)sintheta)/(a)`......(`iii`)
& `T=(pim)/(qB_(0))`.....(`iv`)
where `T` is the time required for the collision to occur
By solving, `v_(0)sintheta costheta=(E_(0))/(B_(0))`
Also, `v_(0)sintheta=(pi)/(2)(E_(0))/(B_(0))`
`rArr costheta=(2)/(pi)` , `theta=cos^(-1)((2)/(pi))`, `v_(0)=(E_(0)//B_(0))/((2)/(pi)sqrt(1-2^(2)//pi^(2)))`
The maximum value of the `Z`-coordinate of the first particle is
`H=(v_(0)^(2)sin^(2)theta)/(2a)=(pi^(2))/(8)(mE_(0))/(qB_(0)^(2))`