MHT CET · Physics · Magnetic Effects of Current

A current 'I' flows in anticlockwise direction in a circular arc of a wire having \(\left(\frac{3}{4}\right)^{\text {th }}\) of circumference of a circle of radius R. The magnetic field ' \(B\) ' at the centre of circle is ( \(\mu_0=\) permeability of free space)

A \(\frac{\mu_0 \mathrm{I}}{3 \mathrm{R}}\) in upward direction
B \(\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}\) in downward direction
C \(\frac{3 \mu_0 \mathrm{I}}{8 \mathrm{R}}\) in downward direction
D \(\frac{3 \mu_0 \mathrm{I}}{8 \mathrm{R}}\) in upward direction

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{3 \mu_0 \mathrm{I}}{8 \mathrm{R}}\) in upward direction

Step-by-step Solution

Detailed explanation

Magnetic field due to \(\theta\) angle \(B=\frac{\theta}{2 \pi} \times \frac{\mu_0 I}{2 R}\)
\(\begin{aligned}
& \theta=\frac{3}{4} \times 2 \pi r=\frac{3}{2} \pi \\
& B=\frac{3 \pi}{4 \pi} \times \frac{\mu_0 I}{2 R}=\frac{3 \mu_0 I}{8 R}
\end{aligned}\)

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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