Magnetic Effects & Magnetism

JEE Main level, single correct, medium

215 of 276

Question 215 of 276

A proton, a deuteron and an α\alpha particle are moving with same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them ______ is and their speed is ______ in the ratio.

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Answer: (A) 2:1:12:1:1 and 4:2:14:2:1

F=q(v⃗×B⃗)=qm(P⃗×B⃗)F=q\left(\vec{v}\times \vec{B}\right)=\dfrac{q}{m}\left(\vec{P}\times \vec{B}\right)
⇒F∝qm\Rightarrow F\propto \dfrac{q}{m}
thus,
F1:F2:F3=q1m1:q2m2:q3m3{F}_{1}:{F}_{2}:{F}_{3}=\dfrac{{q}_{1}}{{m}_{1}}:\dfrac{{q}_{2}}{{m}_{2}}:\dfrac{{q}_{3}}{{m}_{3}}=emp:e2mp:2e4mp=\dfrac{e}{{m}_{p}}:\dfrac{e}{2{m}_{p}}:\dfrac{2e}{4{m}_{p}}=11:12:24=\dfrac{1}{1}:\dfrac{1}{2}:\dfrac{2}{4}=2:1:1=2:1:1
Now for speed calculation, P=constant⇒v∝1mP=constant\Rightarrow v\propto \dfrac{1}{m}
thus v1:v2:v3=1mp:12mp:14mp{v}_{1}:{v}_{2}:{v}_{3}=\dfrac{1}{{m}_{p}}:\dfrac{1}{2{m}_{p}}:\dfrac{1}{4{m}_{p}}=11:12:14=\dfrac{1}{1}:\dfrac{1}{2}:\dfrac{1}{4}=4:2:1=4:2:1.

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