In the XY- plane , the region `y gt 0` has a uniform magnetic field `B_1 hatk` and the region ` y lt 0 `has another uniform magnetic field `B_2 hatk`. A positively charged particle is projected from the origin along the positive y- axis with speed ` v_0= pi m s^(-1)` at t=0 , as shown in the figure . Neglect gravity in this problem . Let t=T be the time when the particle crosses the X- axis from below for the first time . If `B_2 = 4B_1`, the average speed to the particle , in `ms^(-1)`, along the x- axis in the time interval T is __________ .

Answer

1.2

Explanation

(i) Average speed along x - axis

`(v_(x)) = (int|vecx|dt)/(int dt) = (d_(1) + d_(2))/(t_(1) + t_(2))`
(ii) We have,
`r_(1) = (mv)/(qB_(1)), r_(2) = (mv)/(qB_(2))`
Since, `B_(1) = B_(2)/4`
`therefore r_(1) = 4r_(2)`
Time in `B_(1) rarr (xm)/(qB_(1)) = t_(1)`
Time in `B_(2) rArr (pim)/(qB_(2)) = t_(2)`
Total distance along x - axis.
`d_(1) + d_(2) = 2r_(1) - 2r_(2) = 2(r_(1) - r_(2)) = 2(3r_(2)) = 6r_(2)`
Total time `T = t_(1) + t_(2) = 5t_(2)`
`therefore` Average speed `= (6r_(2))/(5t_(2)) = (6mv)/(5qB_(2)) xx (qB_(2))/(pim) = 6/5 = 1.2`