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JEE Advanced 2022 Paper 1 · Physics · Magnetic Effects of Current · hard

A proton accelerated from rest through a potential difference V enters a uniform magnetic field perpendicular to its velocity and moves on a circle…

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A proton accelerated from rest through a potential difference VV enters a uniform magnetic field perpendicular to its velocity and moves on a circle of radius r0r_0. In List-I a different particle (alpha particle: mass 4mp4m_p, charge 2e2e; deuteron: mass 2mp2m_p, charge ee) enters the same field under the stated condition. Match the ratio of its orbit radius to r0r_0 with List-II.

List-I: (P) alpha particle accelerated through the same VV; (Q) deuteron with the same speed as the proton; (R) alpha particle with the same momentum as the proton; (S) alpha particle with the same kinetic energy as the proton.

List-II: (1) 1/21/2; (2) 11; (3) 2\sqrt2; (4) 22; (5) 222\sqrt2.
  1. A. P→2, Q→4, R→1, S→3
  2. B. P→3, Q→4, R→2, S→1
  3. C. P→3, Q→4, R→1, S→2
  4. D. P→3, Q→2, R→1, S→4
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Answer: (C) P→3, Q→4, R→1, S→2

Radius r=pqB=mvqB=2mKqBr=\dfrac{p}{qB}=\dfrac{mv}{qB}=\dfrac{\sqrt{2mK}}{qB}. (P) K=qVK=qV, so r∝m/qr\propto\sqrt{m/q}: ratio (4/2)=2\sqrt{(4/2)}=\sqrt2 (3). (Q) Same speed: r∝m/qr\propto m/q, ratio 22 (4). (R) Same momentum: r∝1/qr\propto1/q, ratio 1/21/2 (1). (S) Same KK: r∝m/qr\propto\sqrt m/q, ratio 4/2=1\sqrt4/2=1 (2). So P→3, Q→4, R→1, S→2.
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