Introduction:
When a current flows through a circular coil, it produces a magnetic field around it. The magnetic field on the axis of a circular coil is of great significance in many applications, such as in the design of magnetic sensors, motors, generators, and other electromagnetic devices. In this response, we will derive an expression for the magnetic field at a point on the axis of a circular coil due to a current flowing in it.

Derivation:
To derive the expression for the magnetic field at a point on the axis of a circular coil, we will use the Biot-Savart law, which states that the magnetic field at a point due to a current-carrying element is directly proportional to the current and the length of the element and inversely proportional to the distance from the point to the element.

Steps:
1. Consider a circular coil of radius R and N turns, carrying a current I.
2. Let P be a point on the axis of the coil at a distance x from the center of the coil.
3. Divide the circular coil into N small current-carrying elements dl, each of length dθ.
4. The magnetic field dB at point P due to a small current-carrying element dl is given by dB = μ0Idl sinθ / (4πr^2), where r is the distance from dl to point P, θ is the angle between the axis of the coil and the line joining dl to P, and μ0 is the permeability of free space.
5. The total magnetic field B at point P due to the entire circular coil is given by B = ∫dB = μ0I(N/2R) ∫sinθ dθ / r^2, where the limits of integration are from 0 to 2π.
6. Simplifying the integral, we get B = μ0IN(x / (x^2 + R^2)^(3/2)), where x is the distance from the center of the coil to the point on the axis and N is the number of turns.

Conclusion:
Thus, the expression for the magnetic field at a point on the axis of a circular coil due to a current flowing in it is B = μ0IN(x / (x^2 + R^2)^(3/2)). This expression helps in the design and analysis of many electromagnetic devices.