Question

Two similar coils A and B of radius 'r' and number of turns 'N' each are placed concentrically with their planes perpendicular to each other. If $ I $ and $ 2I $ are the respective currents passing through the coils then the net magnetic induction at the centre of the coils will be:

• A. \( \sqrt{3} \left( \mu_0 \frac{NI}{2r} \right) \)
• B. \( \sqrt{5} \left( \mu_0 \frac{NI}{2r} \right) \)
• C. \( 5 \mu_0 \frac{NI}{2r} \)
• D. \( 3 \mu_0 \frac{NI}{r} \)

Hint:

When two magnetic fields are perpendicular to each other, their resultant field is given by the Pythagorean theorem. This is a key concept for calculating the net field in such cases.

Answer 1

The Correct Option is B

The magnetic field at the center of a coil due to a current \( I \) is given by: \[ B = \mu_0 \frac{NI}{2r} \] where \( \mu_0 \) is the permeability of free space, \( N \) is the number of turns, \( I \) is the current, and \( r \) is the radius of the coil. For coil A, the magnetic field at the center is: \[ B_A = \mu_0 \frac{NI}{2r} \] For coil B, the current is \( 2I \), so the magnetic field at the center is: \[ B_B = \mu_0 \frac{N(2I)}{2r} = \mu_0 \frac{NI}{r} \] Now, since the coils are placed concentrically and their planes are perpendicular to each other, the net magnetic induction will be the vector sum of the magnetic fields from each coil. Since the angle between the magnetic fields is 90°, we use the Pythagorean theorem to find the resultant: \[ B_{\text{net}} = \sqrt{B_A^2 + B_B^2} \] Substituting the expressions for \( B_A \) and \( B_B \): \[ B_{\text{net}} = \sqrt{\left( \mu_0 \frac{NI}{2r} \right)^2 + \left( \mu_0 \frac{NI}{r} \right)^2} \] \[ B_{\text{net}} = \sqrt{\frac{\mu_0^2 N^2 I^2}{4r^2} + \frac{\mu_0^2 N^2 I^2}{r^2}} \] \[ B_{\text{net}} = \sqrt{\frac{\mu_0^2 N^2 I^2}{4r^2} \left( 1 + 4 \right)} \] \[ B_{\text{net}} = \sqrt{\frac{\mu_0^2 N^2 I^2}{4r^2} \times 5} \] \[ B_{\text{net}} = \sqrt{5} \left( \mu_0 \frac{NI}{2r} \right) \] Thus, the net magnetic induction is \( \sqrt{5} \left( \mu_0 \frac{NI}{2r} \right) \), which corresponds to option (B).