Question

A small square loop of wire, of side l, is placed inside a large square loop of wire of side L (L >> l). The loops are coplanar and their centres coincide. The mutual inductance of the system is proportional to

(1) \(\frac{l}{L}\)

(2) \(\frac{l^2}{L}\)

(3) \(\frac{l}{L}\)

(4) \(\frac{L^2}{l}\)

Answer 1

 (2) \(\frac{l^2}{L}\)

The bigger loop may be taken as primary coil and the smaller one as the secondary coil. (We need to use the expressions for the magnetic field due to a finite wire). The bigger coil can be considered as equivalent to four current carrying long wires. All these wires, it can be seen produce magnetic fields in the same sense. Hence net magnetic field at the centre, will be

The flux, linked with the smaller loop, will be

We can assume the magnetic field to b constant over the whole area of the smaller loop.

From (1) & (2), we get