A wire carrying current has the configuration as shown in the figure. Two semi-infinite straight sections, each tangent to the same circle, are connected by a circular arc, of angle , along the circumference of the circle, with all sections lying in the same plane. We have to find for to be zero at the centre of the circle.

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Let the wire sections carrying current be in the xy plane and the z-axis be to the xy-plane with orientation .

Using the Biot-Savart law we will calculate the magnetic field at the point P by the three sections of the current carrying wire.

We will first calculate the magnetic field at P due to the semi-infinite length section shown in the following figure.

Let us write the field due to the element as at a distance from the open end of the semi-infinite line.

And

We calculate next the integral

.

Making the substitution

we have

Therefore,

Similarly, the magnetic field at P due to the other semi-infinite wire carrying current as shown in the first figure will also be

We next calculate the magnetic field at P due to the circular arc subtending an angle with P.

From the Biot-Savart law

As the radial direction is perpendicular to the tangent at the circumference,

The magnetic field at P due to the arc of angle will be

The magnetic field at P due all the three sections of the wire carrying current will, therefore, be

For to be zero