Question

Two concentric coils, each of radius equal to \(2\pi\) cm are placed at right angle to each other. If \(3\) A and \(4\) A are the currents flowing through the two coils, respectively. The magnetic induction (in Wb/m\({}^{2}\)) at the centre of the coils will be {\(\mu _{0}=4\pi \times 10^{-7}\) SI unit}

• A. \(5\times 10^{-5}\)
• B. \(12\times 10^{-5}\)
• C. \(7\times 10^{-5}\)
• D. \(10^{-5}\)

Hint:

Find the field of each coil at the centre; the two fields are perpendicular, so add them as vectors.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Field at the centre of a circular coil is \(B=\dfrac{\mu_0I}{2r}\), with \(r=2\pi\) cm \(=0.02\pi\) m.

Step 2: Each coil
\(B_1=\dfrac{4\pi\times10^{-7}\times3}{2\times0.02\pi}=3\times10^{-5}\) T and \(B_2=\dfrac{4\pi\times10^{-7}\times4}{2\times0.02\pi}=4\times10^{-5}\) T.

Step 3: Resultant
The coils are at right angles, so the fields are perpendicular: \(B=\sqrt{B_1^2+B_2^2}\).

Step 4: Result
\(B=\sqrt{9+16}\times10^{-5}=5\times10^{-5}\) Wb/m\(^2\).

Final Answer:
The magnetic induction at the centre is 5 x 10^-5 Wb/m squared. \[ \boxed{\text{(A) }5\times10^{-5}} \]