b) For a circular coil of radius R and N turns carrying current I, the magnitude of the magnetic field at a point on its axis at a distance x from its centre is given by,
Consider two parallel co-axial circular coils of equal radius R, and number of turns N, carrying equal currents in the same direction, and separated by a distance R. Show that the field on the axis around the mid-point between the coils is uniform over a distance that is small as compared to R, and is given by,
Let a point P be at a distance of l from the midpoint of the centres of the coils.
The distance of this point from the centre of one coil would be R/2+l and that from the other would be R/2-l.
The magnetic field at P due to one of the coils would be
The magnetic field at P due to the other coil would be
Since the direction of current in both the coils is same the magnetic fields B1 and B2 due to them at point P would be in the same direction
Since l<<R we can ignore term l2/R2
Since the above value is independent of l for small values it is proved that about the midpoint the Magnetic field is uniform.