Two Indentical coils having same number of turns and carrying equal current have common centre and their planes are at right angles to each other. What is the ratio of magnitude of the resultant magnetic field at the centre and magnetic field due to one of the coils at the centre ?

A \(\sqrt{2}: 1\)
B \(1: \sqrt{2}\)
C \(2: 1\)
D \(1: 1\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\sqrt{2}: 1\)

Step-by-step Solution

Detailed explanation

The magnetic field induced by a coil at its centre.
\(B_1=\frac{\mu_0 I}{2 r}=B_2=B\)
Resultant magnetic field produc by two mutually perpendicu coils.
\(\begin{aligned} B _{\text {net }} & =\sqrt{ B _1^2+ B _2^2} \\ \therefore \quad B_{\text {net }} & =\sqrt{ B ^2+ B ^2} \\ \therefore \quad B_{\text {net }} & =\sqrt{2} B \\ \therefore \quad & \frac{ B _{\text {net }}}{ B }=\frac{\sqrt{2}}{1}\end{aligned}\)

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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