Question

Use Biot-Savart law to derive the expression for the magnetic field on the axis of a current carrying circular loop of radius R.
Draw the magnetic field lines due to a circular wire carrying current I.

Answer 1

Biot-Savart Law states that the magnetic field(dB) at point P, due to the small current carrying element Idl of the current carrying conductor is base on three facts.

• directly proportional to the Idl elementof the conductor.
• directly proportional to sinФ.
• inversely proportional to the square of the distance between the point from the current carrying element.
• I am attaching the expression of the current carrying circular loop.

Answer 2

Expression for the magnetic field using Biot-Savart law:

According to Biot-Savart law, the magnetic field at a point P is given as:

dB = μ₀/4π × I (|dl × r|/r³)

Where,

I = Current in the loop

r = Radius of the loop

x = Distance between center point of the loop and point P

dl = Conducting element of the loop

r² = x² + R²

Since dl ⊥ r, then |dl × r| = r.dl

Therefore,

dB = μ₀/4π × I.dl/(x² + R²)

dB is divided into vertical component (dBₓ) and horizontal component (dBₐ).

The vertical component gets cancelled out.

dBₓ = dB cos θ

cos θ = R/(x² + R²$)^{1/2}$

∴ dBₓ = (μ₀ I dI)/4π × R/(x² + R²$)^{3/2}$

∴ $B=B_{x} \hat{i}=\frac{\mu_{0} I R^{2}}{2\left(x^{2}+R^{2}\right)^{1 / 2}} \hat{i}$

The magnetic field lines due to a circular wire are given in the image below.