Any two points on the circumference of uniform circular conductor are connected to the terminals of a cell. Show that, the resultant magnetic field at the centre of the circle is zero.

Explanation

In the figure a circular conductor of radius a is shown.
There are two points, Z,Y on the circumference and O is centre of the circle .
Let `I_(1) and I_(2)` be the currents flowing through the parts XMY and XNY respectively of the circular conductor. Let XMY and XNY subtend angles `theta_(1) and theta_(2)` respectively at the centre.

Now magnetic field intensity at O due to part XMY,
`B_(XMY)=(mu_(0))/(4pi)*(I_(1))/(a) theta_(1)" "....(1)`
ans similarly for the part XNY,
`B_(NXY)=(mu_(0))/(4pi)*(I_(2))/(a) theta_(2) " "....(2)`
`B_(MXY)` is directed into the plane of the conductor, while `B_(XNY)` is directed out of the plane of the conductor.
If r resistance per unit lenght of the conductor pern unit lenght of the conductor then resistance of the part XMY and XNY are given by `R_(1) = (a theta_(1)) r and R_(2)=(a theta_(2))r` resepctively.
As potential differences across `R_(1) and R_(2)` are equal
`:. V_(XMY)=V_(XNY)`
or `I_(1)R_(1)=I_(2)R_(2) or, I_(1)(a theta_(1)) r= I_(2)(a theta_(2))r`
`:. I_(1) theta_(1)=I_(2) theta_(2)`
From equation (1) and (2) if follows that magnetic fields `B_(1) and B_(2)` due to the two parts of the circular conductor are equal and opposite. Hence the magnetic field at the centre of the circular conductor is zero.