The region between `x=0` and `x=L` is filled with uniform steady magnetic field `B_0hatk`. A particle of mass m, positive charge q and velocity `v_0hati` travels along x-axis and enters the region of the magnetic field. Neglect the gravity throughout the question.
a. Find the value of `l` if the particle emerges from the region of magnetic field with its final velocilty at an angel `30^@` to its initial velocity.
b. Find the final velocity of the particle and the time spent by it in the magnetic field, if themagnetic field now extends upto 2.1`L`.

Explanation

To solve the problem step-by-step, we will break it down into parts (a) and (b) as given in the question.

### Part (a)

**Given:**
- A particle of mass \( m \), positive charge \( q \), and initial velocity \( v_0 \hat{i} \) enters a uniform magnetic field \( \vec{B} = B_0 \hat{k} \).
- The particle emerges from the magnetic field at an angle of \( 30^\circ \) to its initial velocity.

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