Two identical coils having radius R and number of turns N are placed co-axially with their centres separated by a distance equal to their radius R. The two coils are given same current I in same direction. The configuration is often known as a pair of Helmholtz coil.
(i) Calculate the magnetic field `(B)` at a point `(P)` on the axis between the coils at a distance x from the centres of one of the coils.
(ii) Prove that `(dB)/(dx) = 0` and `(d^(2)B)/(dx^(2)= 0` [ In fact `(d^(3)B)/( dx^(3))` is also equal to zero ] at the point lying midway between the two coils. What conclusion can you draw from these results?

Answer

`(1) (mu_(0)NIR^(2))/2[1/((R^(2)+x^(2))^(3//2)+1/[R^(2)+(R-x^(2)]^(3//2)]` (2) The magnetic field is constant at points close to mid way between the coils.