In the xy - plane , the region y 0 has a uniform magnetic field B 1 k ^ and the region y 0 has another uniform magnetic field B 2 k ^ . A positively charged particle is projected from the origin along the positive y − axis with speed v 0 = π 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 = 4 B 1 , the average speed of the particle, in m s − 1 , along the x − axis in the time interval T is ________.

Question

In the xy - plane , the region y > 0 has a uniform magnetic field B 1 k ^ and the region y 0 has another uniform magnetic field B 2 k ^ . A positively charged particle is projected from the origin along the positive y − axis with speed v 0 = π 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 = 4 B 1 , the average speed of the particle, in m s − 1 , along the x − axis in the time interval T is ________.,

MARKS App · Accepted Answer

( i ) Average speed along x − axis v x = ∫ v x → d t ∫ d t = d 1 + d 2 t 1 + t 2 ( ii ) We have, r 1 = m v q B 1 , r 2 = m v q B 2 Since, B 1 = B 2 4 ∴ r 1 = 4 r 2 Time in B 1 ⇒ π m q B 1 = t 1 Time in B 2 ⇒ π m q B 2 = t 2 Total distance along x - axis d 1 + d 2 = 2 r 1 + 2 r 2 = 2 r 1 + r 2 = 2 5 r 2 Total time T = t 1 + t 2 = 5 t 2 ∴ Average speed = 10 r 2 5 t 2 = 2 m v q B 2 × q B 2 π m = 2

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