A uniform magnetic field B exists in the region between $$x=0$$ and $$x=3R2$$ (region$$ $$2$$ in the $$figure) pointing normally into the plane of the paper. A particle with charge $$+Q$$ and momentum p directed along $$x-axis$$ enters region $$2$$ from region $$1$$ at point $$P1y=$$−R. Which of the following $$option(s)$$ $$is/are$$ correct?

Question

A uniform magnetic field B exists in the region between $$x=0$$ and $$x=3R2$$ (region$$ $$2$$ in the $$figure) pointing normally into the plane of the paper. A particle with charge $$+Q$$ and momentum p directed along $$x-axis$$ enters region $$2$$ from region $$1$$ at point $$P1y=$$−R. Which of the following $$option(s)$$ $$is/are$$ correct?

Infinity Learn · Accepted Answer

For particle to re $$-$$ enter region $$1$$, radius of circular motion should be less than $$3R2r$$<$$3R2$$⇒mvqB<$$3R2$$⇒B>$$2mv3QR$$⇒B>$$2p3QR$$∴Option (3) is correct Now, for complete semi circular path, Initial Momentum at $$P1$$ $$=pi^$$ Final Momentum at $$P2=pj^$$∴ Change in momentum from $$P1$$ & $$P2=pj^$$−$$pi^$$⇒ Magnitude of Change in momentum $$=p2+p2=2p$$ ∴Option (1) is wrong Distance between the point $$P1$$ and the point of $$re-entry$$ into region $$1=d=2r=2pqB=2mvqB$$⇒d∝m∴Option $$4$$ is $$wrongr1$$−$$cos$$θ$$=R$$−−−−$$1$$rsinθ$$=3R2$$−−−−−−$$2$$$$2$$ ÷$$1$$ we getrsinθ$$r1$$−$$cos$$θ$$=3R2R$$⇒$$2sin$$θ$$2cos$$θ$$22sin2$$θ$$2=32$$⇒$$tan$$θ$$2=23$$−−−−$$3$$$$tan$$θ$$=2tan$$θ$$21-tan2$$θ$$2$$∵$$tan2$$θ$$=2tan$$θ$$1-tan2$$θ $$tan$$θ$$=2231-232=125rsin$$θ$$=3R2$$⇒$$r1213=3R2$$⇒$$r=13R8$$⇒$$13R8=pQB$$⇒$$B=8p13QR$$∴option $$2$$ is correct

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