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MCQEasyJEE 2024Force on Moving Charge

JEE Physics 2024 Question with Solution

A proton and a deuteron (q=+eq = +e, m=2.0um = 2.0u) having the same kinetic energies enter a region of uniform magnetic field BB, moving perpendicular to BB. The ratio of the radius rdr_d of the deuteron path to the radius rpr_p of the proton path is:

  • A

    1:11:1

  • B

    1:21:\sqrt{2}

  • C

    2:1\sqrt{2}:1

  • D

    1:21:2

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: A proton and a deuteron move perpendicular to a uniform magnetic field with the same kinetic energy. For both particles, charge is q=+eq=+e, and for the deuteron md=2mpm_d=2m_p.

Find: The ratio rdrp\frac{r_d}{r_p}.

For motion perpendicular to a magnetic field, the radius of the circular path is

r=mvqBr=\frac{mv}{qB}

Since both particles have the same kinetic energy,

12mpvp2=12mdvd2\frac{1}{2}m_pv_p^2=\frac{1}{2}m_dv_d^2

So,

vp2=mdmpvd2v_p^2=\frac{m_d}{m_p}v_d^2

and therefore

vp=mdmp vdv_p=\sqrt{\frac{m_d}{m_p}}\,v_d

Now,

rp=mpvpqB,rd=mdvdqBr_p=\frac{m_pv_p}{qB}, \qquad r_d=\frac{m_dv_d}{qB}

Hence,

rdrp=mdvdmpvp\frac{r_d}{r_p}=\frac{m_dv_d}{m_pv_p}

Substituting vp=mdmp vdv_p=\sqrt{\frac{m_d}{m_p}}\,v_d,

rdrp=mdvdmp(mdmp vd)=mdmp\frac{r_d}{r_p}=\frac{m_dv_d}{m_p\left(\sqrt{\frac{m_d}{m_p}}\,v_d\right)}=\sqrt{\frac{m_d}{m_p}}

Since md=2mpm_d=2m_p,

rdrp=2\frac{r_d}{r_p}=\sqrt{2}

Therefore, the ratio is 2:1\sqrt{2}:1. The correct option is C.

Using radius dependence on mass at fixed kinetic energy

Given: Both particles have the same kinetic energy and the same charge.

Find: How radius depends on mass.

Using

r=mvqBr=\frac{mv}{qB}

and

K=12mv2K=\frac{1}{2}mv^2

we get

v=2Kmv=\sqrt{\frac{2K}{m}}

So,

r=mqB2Km=2KmqBr=\frac{m}{qB}\sqrt{\frac{2K}{m}}=\frac{\sqrt{2Km}}{qB}

Thus, for fixed KK, qq, and BB,

r∝mr\propto \sqrt{m}

Therefore,

rdrp=mdmp=2\frac{r_d}{r_p}=\sqrt{\frac{m_d}{m_p}}=\sqrt{2}

Hence, the required ratio is 2:1\sqrt{2}:1.

Common mistakes

  • Using r∝mr\propto m directly and forgetting that the velocities are different for equal kinetic energies. Since KK is the same, heavier particles move slower. First relate velocity to mass using kinetic energy, then compare radii.

  • Assuming the deuteron and proton have the same velocity because they enter the same magnetic field. The field is common, but the question states equal kinetic energies, not equal speeds. Use 12mv2\frac{1}{2}mv^2 carefully.

  • Not cancelling the charge correctly. Both particles have charge +e+e, so qq cancels in the ratio. The difference comes only from mass and the resulting speed.

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