A particle having mass `m` and charge `q` is released from the origin in a region in which electric field and megnetic field are given by `vec(B) = -B_(0)hat(j)` and `vec(E) = vec(E)_(0)hat(k)`. Find the speed of the particle as a function of its `z`-coordinate.

Explanation

The magnetic force changes direction of velocity and not its magnitude. The electric force definitely changes velocity, either magnitude or both magnitude and direction. Work is done only by electric force
Let particle be displaced along z-axis by `z`
Work done by electric force
`W = (q E_(0) hat(k)).(z hat(k))`
`= q E_(0) z`
Work done by work energy theorem
`W = Delta K`
`qE_(0)z = (1)/(2) mv^(2) - 0`
`v = sqrt((2qE_(0) z)/(m))`
Note: Magnetic force rotates the particle in plane `_|_^(ar)` to `vec(B)` axis.