# Solved Examples : Magnetic field due to circular loop

Example : In the figure shown a current 2i is flowing in a straight conductor and entering along the diameter of the circular loop of similar conductor through the point A. The current is leaving the loop along another similar semi-infinite conductor parallel to the plane of the loop through the other opposite end D of the diameter.

What is the magnetic field at the centre of the loop P?

Solution : The magnetic field at the centre P due to the entering current along diameter is zero.

The magnetic field at P due to the semicircular loop AED is

$\displaystyle \vec{B_1} = \frac{\mu_0}{4\pi} \frac{i}{a}(\pi) \hat{k}$

The magnetic field at P due to the semi-circular segment AGD is

$\displaystyle \vec{B_2} = \frac{\mu_0}{4\pi} \frac{i}{a}(\pi) (-\hat{k})$

and the field at P due to the semi infinite straight conductor is given by

$\displaystyle \vec{B_3} = \frac{\mu_0}{4\pi} \frac{2 i}{a} (-\hat{k})$

The net field at P is

$\displaystyle \vec{B} = \vec{B_1} + \vec{B_2} + \vec{B_3}$

$\displaystyle \vec{B} = \frac{\mu_0}{4\pi} \frac{2 i}{a} (-\hat{k})$

The net magnetic field at P is along the perpendicular to the plane of the loop downward.

Example : An infinitely long conductor carrying a current i with a semicircular loop in the x-y plane and two straight parts, one parallel to x-axis and another coinciding with z-axis is placed. What is the magnetic field at the centre P of the semicircular loop?

Solution : The magnetic field at P due to the straight part of conductor parallel to x-axis is

$\displaystyle \vec{B_1} = \frac{\mu_0}{4\pi} \frac{i}{d/2} (\hat{k})$

The magnetic field at the point P due to the semicircular loop on x – y plane is given by,

$\displaystyle \vec{B_2} = \frac{\mu_0}{4\pi} \frac{\pi i}{d/2} (\hat{k})$

The magnetic field at P due to the straight part coinciding with z-axis is given by,

$\displaystyle \vec{B_3} = \frac{\mu_0}{4\pi} \frac{i}{d/2} (-\hat{i})$

The total magnetic field at P is given by

$\displaystyle \vec{B} = \vec{B_1} + \vec{B_2} + \vec{B_3}$

$\displaystyle \vec{B_3} = \frac{\mu_0}{4\pi} \frac{2 i}{d} [(1+\pi)\hat{k} – \hat{i}]$

Exercise 2: Find the magnetic field at the origin due to the combination of two semi infinite wires and a semicircular wire as shown.

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