The region between ` x=o and x = L` is filled with uniform, steady magnetic field `B_(0) hat(k)`. A particle of mass `m` , positive charge `q` and velocity `v_(0) hat(i)` travels along ` x-axis` and enters the region of the magnetic field. Neglect gravity throughout the question .
(a) Find the value of `L` if the particle emerges from the region of magnetic field with its final velocity at angle `30^(@)` to its initial velocity.
(b) Find the velocity of the particle and the time spent by it in the magnetic field, if the magnetic field now extends upto `2.1L`.

Explanation

(a)
Radius of circular path
`R = (mv_(0))/(B_(0)q)`
Angle of deviation
`sin theta = (L)/(R )`
`sin 30^(@) = L//R`
`L = (R )/(2) = (mv_(0))/(2B_(0)q)`
(b) Now width of magnetic field is
`2.1 L = (2.1 mv_(0))/(2B_(0)q) gt (mv_(0))/(B_(0)q) (= R)`
The particle enters and emerges from same side

Final velocity `vec(v') = -v_(0)hat(i)`
Angle of deviation `= pi`
Time to cross magentic field
`t = (m theta)/(B_(0)q) = (m pi)/(B_(0)q)`