Question

Two similar coils of radius $R$ are lying concentrically with their planes at right angles to each other. The currents flowing in them are $I$ and $2I$, respectively. The resultant magnetic field induction at the centre will be

• A. $\frac{\sqrt5 \mu_0 I}{2R}$
• B. $\frac{\sqrt \mu_0 I}{2R}$
• C. $\frac{\mu_0 I}{2R}$
• D. $\frac{\mu_0 I}{R}$

Answer 1

The Correct Option is A

To solve this problem, we need to determine the resultant magnetic field at the center due to the two concentric coils.

The magnetic field B at the center of a single circular coil carrying current I and having radius R is given by the formula:

B = \frac{\mu_0 I}{2R}

Here, \mu_0 is the permeability of free space.

Let's calculate the magnetic fields due to each coil:

1. For the first coil with current I:

   B_1 = \frac{\mu_0 I}{2R}

2. For the second coil with current 2I:

   B_2 = \frac{\mu_0 (2I)}{2R} = \frac{\mu_0 I}{R}

Since the coils are perpendicular to each other, the magnetic fields B_1 and B_2 are perpendicular. The magnitude of the resultant magnetic field B_{\text{resultant}} at the center is given by applying the Pythagorean theorem:

B_{\text{resultant}} = \sqrt{B_1^2 + B_2^2}

Substitute the values:

B_{\text{resultant}} = \sqrt{\left(\frac{\mu_0 I}{2R}\right)^2 + \left(\frac{\mu_0 I}{R}\right)^2}

B_{\text{resultant}} = \sqrt{\frac{\mu_0^2 I^2}{4R^2} + \frac{\mu_0^2 I^2}{R^2}}

Simplifying:

B_{\text{resultant}} = \sqrt{\frac{\mu_0^2 I^2}{4R^2} + \frac{4\mu_0^2 I^2}{4R^2}}

B_{\text{resultant}} = \sqrt{\frac{5\mu_0^2 I^2}{4R^2}}

Finally:

B_{\text{resultant}} = \frac{\sqrt{5} \mu_0 I}{2R}

Thus, the correct answer is \frac{\sqrt{5} \mu_0 I}{2R}.