Question

Two identical current carrying coils with same centre are placed with their planes perpendicular to each other. If i = \(\sqrt{2}A\) and radius of coil R = 1 m, then magnetic field at centre C is equal to:

• A. \(\mu_{0}\)
• B. \(\frac{\mu_{0}}{2}\)
• C. \(2\mu_{0}\)
• D. \(\sqrt{2}\mu_{0}\)

Answer 1

The Correct Option is A

The problem involves two identical current-carrying coils placed with their planes perpendicular to each other. We are required to find the magnetic field at the center, C.

Given data:

• Current, \(i = \sqrt{2} \, \text{A}\)
• Radius of coil, \(R = 1 \, \text{m}\)

To find the magnetic field at the center of each coil, we use the formula for the magnetic field at the center of a circular current loop:

\(B = \frac{\mu_0 i}{2R}\)

Substituting the given values:

\(B_1 = \frac{\mu_0 \cdot \sqrt{2}}{2 \cdot 1} = \frac{\mu_0 \sqrt{2}}{2}\)

Since the fields due to the two coils are perpendicular, we use the Pythagorean theorem to find the resultant magnetic field \(B_r\) at the center:

\(B_r = \sqrt{B_1^2 + B_2^2}\)

Where both \(B_1\) and \(B_2\) are equal.

Substituting, we get:

\(B_r = \sqrt{\left(\frac{\mu_0 \sqrt{2}}{2}\right)^2 + \left(\frac{\mu_0 \sqrt{2}}{2}\right)^2}\)

Calculating further:

\(B_r = \sqrt{\frac{\mu_0^2 \cdot 2}{4} + \frac{\mu_0^2 \cdot 2}{4}}\)

\(B_r = \sqrt{\frac{\mu_0^2 \cdot 4}{4}}\)

\(B_r = \sqrt{\mu_0^2}\)

\(B_r = \mu_0\)

Therefore, the resultant magnetic field at the center is \(\mu_0\), which matches the correct option.