Question

Derive an expression of intensity of magnetic field at the centre of a current-carrying circular coil at its centre. Also, enunciate the law used in it.

Hint:

The magnetic field at the center of a current-carrying coil is directly proportional to the current and inversely proportional to the radius of the coil.

Answer 1

Step 1: Biot-Savart Law.
The magnetic field at the center of a current-carrying circular coil is derived using the Biot-Savart law. The Biot-Savart law states that the magnetic field \( d\vec{B} \) at a point due to an infinitesimal current element is: \[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} \] where: - \( \mu_0 \) is the permeability of free space, - \( I \) is the current, - \( d\vec{l} \) is the infinitesimal length element of the wire, - \( \hat{r} \) is the unit vector from the current element to the point where the magnetic field is being calculated, - \( r \) is the distance from the current element to the point.
Step 2: Magnetic field at the center of the coil.
For a circular loop of radius \( R \), the magnetic field at the center of the loop due to a current \( I \) is given by: \[ B = \frac{\mu_0 I}{2R} \] where: - \( B \) is the magnetic field at the center of the coil, - \( \mu_0 \) is the permeability of free space, - \( I \) is the current flowing through the coil, - \( R \) is the radius of the coil.
Step 3: Conclusion.
Thus, the intensity of the magnetic field at the center of a current-carrying circular coil is: \[ B = \frac{\mu_0 I}{2R} \]
Step 4: Law used.
The law used to derive this expression is the Biot-Savart law.