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

As the initial velocity of the particle is perpendicular to the field, the particle will move along the arc of a circle as shown If r is the radius of the circle, then
`(mv_0^2)/r=qv_0B_0 implies r=(mv_0)/(qB_0)`

Also, from geometry, `L=r sin30^@impliesr=2L`
or `L=r/2=(mv_0)/(2qB_0)`
(b) In this case, `x=(2.1mv_0)/(2qB_0)gtr`
Hence, the particle will complete semicircle path and emerge from the field with velocity `-v_0hati` as shown in Fig. Time spent by the particle in the magnetic field `T=(pir)/(v_0)=(pim)/(qB_0)`
The speed of the particle does not change due to magnetic field.
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