There exists a uniform magnetic field `vec(B) = +B_(0) hat(k) " for " x gt 0 and vec(B) = 0` for all `x lt 0`. A charged particle placed at `(0, -a,0)` is given an initial velocity `vec(v) = v_(0) hat(i)`. What is the magnitude and nature of charge on the particle such that it crosses through the origin ?
Strategy: When the charge is projected perpendicular to a uniform magnetic field, it follows a circular path. In this case, the force acting on it will be directed either towards +y-axis. or y-axis. It is given that it crosses point O. Thus, the force at `(0, -a, 0)` must be towards y-axis

Explanation

`vec(F) = q (vec(v) xx vec(B))`
`rArr vec(F) = q(v_(0) hat(i) xx B_(0) hat(k)) = -q v_(0) B hat(j)`
`rArr` q must be negative
Let C be the centre of the circle
such that `R = (a)/(2)`
`(mv_(0))/(|q|B_(0)) = (a)/(2)`
`:. q= - (2mv_(0))/(B_(0)a)`