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)` As the magnetic field `vec(B)` is in `x-y` plane and subtends an angle of `45^(@)` with the x axis
Hence, `B_(x)=B cos 45^(@)=B sqrt(2)` and
`B_(y)B sin 45^(@)=B//sqrt(2)`
So in vector form `" "vec(B)=hat(i)((B)/(sqrt(2)))+hat(j)((B)/(sqrt(2)))`
And `vec(M)=I_(0)hat(k)=I_(0)Shat(k)L^(2)hat(k)`
So, `vec(tau)=MxxB=I_(0)^(2)hat(k)xx((B)/(sqrt(2))hat(i)+(B)/(sqrt(2))hat(j))`
`i.e., vec(tau)=(I_(0)L^(@)B)/(sqrt(2))(hat(j)-hat(i))`
`i.e.` torque has magnitude `I_(0)^(2)L^(2)B` and is directed along line `QS` from `Q` to `S` moment of ineria of the frame about `QS`.