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

2 or 1.2

Explanation

`R_(1)=(mv_(0))/(qB_(1)), t_(1)=(pim)/(qB_(1))`
`R_(2)=(mv_(0))/(qB_(2)), t_(2)=(pim)/(qB_(2))`
Case 1:

Magnitude of average velocity along x-axis
`=(2R_(1)-2R_(2))/(t_(1)+t_(2))=(2(mv_(0))/q(1/(B_(1))-1/(B_(2))))/((mupi)/q(1/(B_(1))+1/(B_(2))))=2((pi)(1/(B_(1))-1/(4B_(1))))/((1/(B_(1))+1/(4B_(1))))=2(3/5)=1.2`
Case 2:
Distance along x-axis `=2R_(1)+2R_(2)=2(mv_(0))/q(1/(B_(1))+1/(B_(2)))`
Time `=t_(1)+t_(2)=(pim)/(qB_(1))+(pim)/(qB_(2))` , Average speed `-((2mv_(0))/q(1/(B_(1))+1/(B_(2))))/((pim)/q(1/(B_(1))+1/(B_(2))))=2/(pi)v_(0)=2`