Q.

In the xy-plane, the region y > 0 has a uniform magnetic field B1k^ and the region y < 0 has another  uniform magnetic field B2k^ . A positively charged particle is projected from the origin along the positive y-axis with speed v0=πm/s 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  B2=4B1, the average speed of the particle, in  m/s, along the x-axis in the time interval T is___________.

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Detailed Solution

1)Average speed along x-axis Vx=∫V¯xdt∫dt=d1+d2t1+t2                        2)We have  r1=mvqB1,r2=mvqB2Since B1=B24∴r1=4r2                       Time in  B1⇒πmqB1=t1∵For full revolution T=2πmqBTime in B2⇒πmqB2=t2Total distance along x-axis  d1+d2=2r1+2r2=2r1+r2=25r2Total time  T=t1+t2=5t2            ∴Average speed  =10r25t2=2mvqB2×qB2πm=2vπ=2ππ=2∵given v=π
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