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A uniform magnetic field vecB is set up along the positive x-axis. A particle of charge q and mass m moving with a velocity vecv enters the field at the origin in X - Y plane such that it has velocity components both along and perpendicular to the magnetic field vecB. Trace, giving reasons, the trajectory followed by the particle. Find out the expression for the distance moved by the particle along the magnetic field in one rotation. |
| Answer» <html><body><p></p>Solution :<a href="https://interviewquestions.tuteehub.com/tag/consider-2017521" style="font-weight:bold;" target="_blank" title="Click to know more about CONSIDER">CONSIDER</a> a charged particle of mass m and charge q entering a uniform magnetic field `vecB` with a <a href="https://interviewquestions.tuteehub.com/tag/velocity-1444512" style="font-weight:bold;" target="_blank" title="Click to know more about VELOCITY">VELOCITY</a> `vecv` along an angle `theta` with the direction of magnetic field in the plane of peper . <br/> Velocity `vecv` may be resolved into two components(i) `v cos theta` along the magnetic field, and (ii) `v sin theta` perpendicular to the magnetic field as shown in fig. Obviously due to normal <a href="https://interviewquestions.tuteehub.com/tag/component-926634" style="font-weight:bold;" target="_blank" title="Click to know more about COMPONENT">COMPONENT</a> `v sin theta` the charged particle will ecperience a force `F = q(v sin theta) B` along a direction perpendicular to `vecB` as well as `(v sin theta)` i.e, force F is along the z-axis. Under its effect, the charged particle describe a <a href="https://interviewquestions.tuteehub.com/tag/circular-916697" style="font-weight:bold;" target="_blank" title="Click to know more about CIRCULAR">CIRCULAR</a> path of radius r, such that <br/> `B q v sin theta = (m (v sin theta)^2)/(r)` <br/> `implies r = (m v sin theta)/(B q) "" ......(i)` <br/> The time period of revolution will be given by <br/> `T = (2 pi r)/((v sin theta)) = (2 pi m)/(B q) "" .......(ii)` <br/> or Frequency of revolution `v = 1/T = (Bq)/(2 pi m) ""........(iii)` <br/> Due to component `v cos theta` the charged particle does not experience force due to magnetic field and tends to move linearly with a constant speed. Thus, under the combined effect of both the velocity componetns we can say that the charged particlewill describe a helical path having its axis parallel to magnetic field. <br/> In one <a href="https://interviewquestions.tuteehub.com/tag/complete-423576" style="font-weight:bold;" target="_blank" title="Click to know more about COMPLETE">COMPLETE</a> revolution of its circular path the charged particle covers a linear distance along `vecB`, which is called as "pitch" of helix and is denoted by p. Obviously <br/> `p = (v cos theta) T = (2 pi m v cos theta)/(Bq)`<br/> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/U_LIK_SP_PHY_XII_C04_E10_010_S01.png" width="80%"/></body></html> | |