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.