**Derivation of Magnetic Field due to a Current Carrying Coil**

**Introduction:**
When a current flows through a coil, it produces a magnetic field around it. The magnetic field can be calculated using Ampere's law, which relates the magnetic field to the current enclosed by a closed loop.

**Derivation:**
Consider a circular coil of radius r carrying a current I. We need to find the magnetic field at a point P located at a distance x from the center of the coil along the X-axis.

**Step 1:**
Divide the circular coil into a large number of small current elements ΔI. Each small current element will produce a magnetic field at point P.

**Step 2:**
Consider a small current element ΔI located at a point A on the coil. The magnetic field produced by this element at point P can be calculated using the Biot-Savart law.

The Biot-Savart law states that the magnetic field dB produced by a small current element ΔI at a point P is given by:
dB = (μ₀/4π) * (ΔI * sinθ) / r²

Where,
μ₀ is the permeability of free space (4π x 10^-7 Tm/A),
θ is the angle between the current element and the line joining the current element to point P,
r is the distance between the current element and point P.

**Step 3:**
The angle θ can be calculated using trigonometry. Since we are considering a circular coil, the angle θ will be the same for all the current elements.

θ = sin^(-1)(x / r)

**Step 4:**
The magnetic field dB produced by the small current element ΔI at point P can be written as:
dB = (μ₀/4π) * (ΔI * sinθ) / r²
= (μ₀/4π) * (ΔI * (x / r)) / r²
= (μ₀/4π) * (ΔI * x) / r³

**Step 5:**
To find the total magnetic field B at point P, we need to consider the contributions from all the small current elements by integrating the expression for dB over the entire coil.

B = ∫(μ₀/4π) * (ΔI * x) / r³

**Step 6:**
Since we are considering a circular coil, the current element ΔI can be written in terms of the total current I and the number of turns N of the coil.

ΔI = I / N

Substituting this into the expression for B, we get:

B = ∫(μ₀/4π) * (I * x) / (N * r³)

**Step 7:**
The integral can be simplified by considering the symmetry of the coil. By symmetry, we know that the magnetic field due to all the current elements on one side of the coil will cancel out the magnetic field due to the current elements on the other side.

Therefore, we only need to consider the contributions from the current elements on one side of the coil.

**Step 8:**
Assuming the coil is tightly wound with a large number of turns N, we can approximate the integral as a summation over all the current elements.

B = (μ₀/4π)