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Question:

(a) State Ampere's circuital law, expressing it in the integral form. (b) Two long coaxial insulated solenoids, S1 and S2 of equal lengths are wound one over the other. A steady current "I" flows through the inner solenoid S1 to the other end B, which is connected to the outer solenoid S2 through which the same current "I" flows in the opposite direction so as to come out at end A. If n1 and n2 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.

Solution:

(a)Ampere's circuital Law states that the circulation of the resultant magnetic field along a closed, plane curve is equal to μ₀ times the total current crossing the area bounded by the curve. ∮B⋅dl = μ₀Ienclosed
(b)(i) Inside a solenoid, magnetic field is given by μ₀nI. In the given scenario, B = μ₀I(n₁ - n₂) as the currents are in the opposite direction. Direction of magnetic field is parallel to the axis of the solenoid and depends on the sign of n₁ - n₂.
(ii) Outside a solenoid, magnetic field is zero.