Two long coaxial insulated solenoids, `S_1 and S_2` of equal lengths are wound one over the other as shown in the figure. A steady current "I" flow through the inner solenoid `S_1` to the other end B, which is connected to the outer solenoid `S_2` through which the same current "I" flows inthe opposite direction so as to come out at end A. If `n_1` and `n_2` are the number of turns per unit length, find the magnitude and direction of the net magnetic field at a point (i) inside on the axis and (ii) outside the combined system.

Explanation

(i) Inside the axis - Field due to solenoid `S_1, B_1 = mu_0 n_1 I` and field due to solenoid `S_2, B_2 = mu_0 n_2I`
Moreover, directions of `vec(B_1) and vec(B_2)` are mutually opposite, hence net magnetic field
`B = B_1 - B_2 = mu_0 (n_1 - n_2) I` in the direction of `vec(B_1)`
(ii) Outside the combined system - Field due to both the solenoids is zero. Hence, net field is zero.