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The region between x = 0 and x = L is filled with uniform, steady magnetic field B 0 . A particle of mass m, positive 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 final velocity of the particle and the time spent by it in the magnetic field, if the magnetic field now extends up to 2.1 L.

The region between x = 0 and x = L is filled with uniform, steady magnetic field B0 . A particle of mass m, positive 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 final velocity of the particle and the time spent by it in the magnetic field, if the magnetic field now extends up to 2.1 L.

Grade:upto college level

1 Answers

Deepak Patra
askIITians Faculty 471 Points
7 years ago
Hello Student,
Please find the answer to your question
KEY CONCEPT : This question involyes a simple understanding of the motion of charged particle in a magnetic field.
Let the particle emerge out from the region of magnetic field at point P. Then the velocity vector 0 makes an angle 30° with x-axis. The normal to circular path at P intersects the negative y – axis at point A.
Hence, AO = AP = R = radius of circular path, which can be found as
Mv20 / R = B0 qv0 ⇒ R = mv0 / qB0 ….(i)
In ∆APM, R sin 30° = L ⇒ R / 2 = L …(ii)
From (i) and (ii), L = mv0 / 2qB0
As the new region of magnetic field is 2.1 L
= 2.1 R / 2 which is obviously > R.
Thus, the required velocity = - v0 .
Since the time period for complete revolution = 2πm / qB0 The time taken by the particle to cross the region of magnetic field = πm / qB0.
Thanks
Deepak patra
askIITians Faculty

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