Question

Derive the formula for the intensity of magnetic field produced at the centre of a current carrying circular loop.

Hint:

The magnetic field at the centre of a current-carrying loop depends on the current and the radius of the loop. It is directly proportional to the current and inversely proportional to the radius.

Answer 1

Step 1: Formula for Magnetic Field at the Centre of a Circular Loop.
The magnetic field \( B \) at the centre of a current-carrying circular loop is given by Ampere’s Law: \[ B = \frac{\mu_0 I}{2R} \] Where:
- \( B \) is the magnetic field,
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{}^2 \) is the permeability of free space,
- \( I \) is the current flowing through the loop,
- \( R \) is the radius of the circular loop.
Step 2: Derivation.
Using the Biot-Savart law for a current-carrying element, the magnetic field produced by a small segment of current is integrated over the entire loop. The result of this integration gives the above formula for the magnetic field at the centre of the loop.
Final Answer:
The formula for the magnetic field at the centre of a current-carrying circular loop is: \[ B = \frac{\mu_0 I}{2R} \]