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Wave Optics

Question
CBSEENPH12039524

Proton, Deuteron and alpha particle of the same kinetic energy is moving in circular trajectories in a constant magnetic field. The radii of the proton, deuteron and alpha particle are respectively rp, rd and rα. Which one of the following relations is correct?

  •  rα = rp= rd

  •  rα = rp< rd

  •  rα > rd> rp

  •  rα = rd> rp

Solution

B.

 rα = rp< rd

For charged particle moving with a speed v, in magnetic field B, on a circular track of radius
straight r space equals space mv over qB space equals space fraction numerator square root of 2 km end root over denominator qB end fraction
open parentheses therefore space mv space equals space straight p space and space straight K space equals fraction numerator straight p squared over denominator 2 straight m end fraction space rightwards double arrow space straight p space equals space square root of 2 km end root close parentheses
rightwards double arrow space straight r space proportional to space fraction numerator square root of straight m over denominator straight q end fraction
or space straight m subscript straight d space equals 2 straight m subscript straight p space and space straight q subscript straight d space equals space straight q subscript straight p semicolon
straight m subscript straight alpha space equals space 4 straight m subscript straight p space and space straight q subscript straight alpha space equals space 2 straight q subscript straight p
rightwards double arrow straight r subscript straight p colon straight r subscript straight d colon straight r subscript straight alpha space equals space fraction numerator square root of straight m subscript straight p end root over denominator straight q subscript straight p end fraction colon fraction numerator square root of 2 straight m subscript straight p end root over denominator straight q subscript straight p end fraction colon fraction numerator square root of 4 straight m subscript straight p end root over denominator 2 straight q subscript straight p end fraction
space equals space 1 colon space square root of 2 colon 1
straight r subscript straight alpha space equals space straight r subscript straight p space less than thin space straight r subscript straight d

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