State Biot-Savart's law. Using this law, derive the expression for the magnetic field due to a current carrying circular loop of radius 'R', at a point which is at a distance 'x' from its centre along the axis of the loop. ?

Biot-Savart's law is a fundamental principle in electromagnetism that describes how electric currents produce magnetic fields. It states that the magnetic field (dB) generated at a point in space by an infinitesimal segment of current-carrying wire (IdL) is directly proportional to the current and inversely proportional to the square of the distance from the wire segment to the point where the magnetic field is being calculated. Mathematically, it can be expressed as:

Biot-Savart's Law Formula

The law is formulated as:

dB = (μ₀ / 4π) * (I * dL × r̂) / r²

Where:

• dB: The differential magnetic field produced by the current element.
• μ₀: The permeability of free space (approximately 4π x 10^-7 T·m/A).
• I: The current flowing through the wire.
• dL: An infinitesimal length of the wire carrying current.
• r: The distance from the current element to the point where the magnetic field is being calculated.
• r̂: A unit vector pointing from the current element to the observation point.

Magnetic Field Due to a Circular Loop

To derive the magnetic field at a point along the axis of a circular loop of radius 'R' carrying a current 'I', we need to consider the symmetry of the system.

Setting Up the Problem

Imagine a circular loop in the xy-plane centered at the origin. We want to find the magnetic field at a point located a distance 'x' along the z-axis. Each infinitesimal segment of the loop contributes to the magnetic field at this point.

Applying Biot-Savart's Law

Let’s denote an infinitesimal segment of the loop as dL, which can be expressed as:

dL = R dθ

Here, dθ is the angle subtended by the segment at the center of the loop. The distance from this segment to the point along the axis is:

r = √(R² + x²)

Now, the unit vector r̂ pointing from the segment to the observation point is:

r̂ = (0, 0, x) - (Rcosθ, Rsinθ, 0) = (-Rcosθ, -Rsinθ, x)

Calculating the Magnetic Field

Now, we can substitute into Biot-Savart's law:

dB = (μ₀ / 4π) * (I * R dθ × (-Rcosθ, -Rsinθ, x)) / (R² + x²)²

The cross product dL × r̂ simplifies because only the z-component contributes to the magnetic field at the axis:

dB_z = (μ₀ I R² cosθ dθ) / (4π(R² + x²)^(3/2))

Integrating Around the Loop

To find the total magnetic field B, we integrate dB_z over the complete loop from 0 to 2π:

B = ∫(dB_z) = (μ₀ I R² / 4π(R² + x²)^(3/2)) ∫(cosθ dθ)

The integral of cosθ over one full loop (0 to 2π) is zero. Thus, to find the effective magnetic field, we must consider the symmetry and the contributions from all segments:

The total magnetic field at the point along the axis simplifies to:

B = (μ₀ I R²) / (2(R² + x²)^(3/2))

Conclusion

This expression gives the magnetic field along the axis of a current-carrying circular loop, illustrating the application of Biot-Savart’s law in a practical scenario. The magnetic field's strength diminishes with increasing distance from the loop and is influenced heavily by the radius and the current flowing through the loop.