MHT CET · Physics · Magnetic Effects of Current

Electron of mass ' \(\mathrm{m}\) ' and charge ' \(\mathrm{q}\) ' is travelling with speed ' \(v\) ' along a circular path of radius ' \(R\) ', at right angles to a uniform magnetic field of intensity ' \(\mathrm{B}\) '. If the speed of the electron is halved and the magnetic field is doubled, the resulting path would have radius

A \(4 \mathrm{R}\)
B \(2 \mathrm{R}\)
C \(\frac{\mathrm{R}}{2}\)
D \(\frac{\mathrm{R}}{4}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{\mathrm{R}}{4}\)

Step-by-step Solution

Detailed explanation

From cyclotron motion and uniform circular motion,
i.e., \(\mathrm{q} v \mathrm{vB} \sin 90^{\circ}=\frac{m v^2}{\mathrm{R}}\)
\(\therefore \quad \mathrm{R}=\frac{\mathrm{mv}}{\mathrm{qB}}\)
Given: \(\mathrm{v}^{\prime}=\frac{\mathrm{v}}{2}\) and \(\mathrm{B}^{\prime}=2 \mathrm{~B}\)
\(\begin{aligned}
\therefore \quad \mathrm{R}^{\prime} & =\frac{\mathrm{mv}}{\mathrm{qB}^{\prime}} \\
& =\frac{\mathrm{m} \frac{\mathrm{v}}{2}}{\mathrm{q} 2 \mathrm{~B}}=\frac{1}{4} \frac{\mathrm{mv}}{\mathrm{qB}} \\
\Rightarrow \mathrm{R}^{\prime} & =\frac{1}{4} \mathrm{R}
\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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