MHT CET · Physics · Magnetic Effects of Current

The ratio of magnetic field at the centre of the current carrying circular loop and magnetic moment is ' \(x\) '. When both the current and radius are double then the ratio will be

A \(2 \mathrm{x}\)
B \(\frac{x}{2}\)
C \(\frac{x}{4}\)
D \(\frac{x}{8}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{x}{8}\)

Step-by-step Solution

Detailed explanation

The magnetic field at the center of the wire is given as \(B=\frac{\mu_0 i}{2 \pi r}\).
The magnetic moment is given as \(\mathrm{M}=\mathrm{i} \pi \mathrm{r}^2\)
The ratio is \(\frac{\frac{\mu_0 \mathrm{i}}{2 \pi \mathrm{r}}}{\mathrm{i} \pi \mathrm{r}^2}=x\) ....(i)
\(x=\frac{\mu_0}{2 \pi^2 r^3}\)
When current and radius is doubled,
\(x_1=\frac{\frac{\mu_0 2 i}{2 \pi 2 r}}{2 i \pi(2 r)^2}\) ....(From (i))
\(x_1=\frac{\mu_0}{16 \pi^2 r^3}\)
\(\mathrm{x}_1=\frac{1}{8}\left(\frac{\mu_0}{2 \pi^2 \mathrm{r}^3}\right)\)
\(\therefore \quad x_1=\frac{x}{8}\)

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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