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.

Also Read:

→ Biot-Savart’s Law
→Magnetic field due to straight conductor carrying current
→ Magnetic field due to Circular Loop
→ Magnetic field at the axis of Circular Loop
→ Ampere’s Circuital Law & its Applications
→ Magnetic field on the axis of a long solenoid
→ Motion of charged particle in a magnetic field
→ Deviation of charged particle in uniform magnetic field & Cyclotron
→ Force on a current carrying wire in a magnetic field
→ Force between two parallel current carrying wires
→Torque on a current carrying loop in a uniform magnetic field
→ Moving coil Galvanometer

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