Using Biot-Savart’s law, derive an expression for magnetic field at any point on axial line of a current carrying circular loop. Hence, find magnitude of magnetic field intensity at the centre of circular coil.

To derive the expression for the magnetic field at any point on the axial line of a current-carrying circular loop using Biot-Savart’s law, we start with the law itself, which states that the magnetic field \( \mathbf{B} \) due to a small segment of current \( I \, d\mathbf{l} \) at a point in space is given by:

Biot-Savart's Law

The formula is expressed as:

dB = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3}

where \( \mu_0 \) is the permeability of free space, \( \mathbf{r} \) is the position vector from the current element to the point where the field is being calculated, and \( r \) is the distance from the current element to that point.

Setup for the Circular Loop

Consider a circular loop of radius \( R \) carrying a current \( I \). We want to find the magnetic field at a point along the axis of the loop, at a distance \( z \) from the center of the loop.

Calculating the Magnetic Field

• Each small segment \( d\mathbf{l} \) of the loop contributes to the magnetic field at the point on the axis.
• The distance from the segment to the point is \( r = \sqrt{R^2 + z^2} \).
• The angle \( \theta \) between \( d\mathbf{l} \) and \( \mathbf{r} \) is such that \( \sin \theta = \frac{R}{r} \) and \( \cos \theta = \frac{z}{r} \).

Using symmetry, the vertical components of the magnetic field from opposite segments cancel out, leaving only the axial components. Thus, the total magnetic field \( B \) at the point on the axis is:

B = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}

Magnitude at the Center of the Coil

To find the magnetic field intensity at the center of the circular coil, we set \( z = 0 \). Substituting this into our expression gives:

B_{center} = \frac{\mu_0 I}{2R}

This result shows that the magnetic field at the center of a circular coil is directly proportional to the current and inversely proportional to the radius of the coil.