A small solenoid (radius rₐ ) is inside a larger solenoid (radius r₆ > rₐ ). They are coaxial with nₐ and n₆ turns per unit length, respectively. The solenoids carry the same current, but in opposite directions. Let r be the radial distance from the common axis of the solenoids. If the magnetic field inside the inner solenoid (r < rₐ) is to be in the opposite direction as the field between the solenoids (rₐ < r < r₆) , but have half the magnitude, determine the required ratio n₆/nₐ .

Verified step by step guidance

1. Step 1: Understand the problem setup. We have two coaxial solenoids, one inside the other, with the inner solenoid having a smaller radius (rₐ) and the outer solenoid having a larger radius (r₆). Both solenoids have different numbers of turns per unit length, nₐ for the inner and n₆ for the outer, and carry the same current in opposite directions.
2. Step 2: Recall the formula for the magnetic field (B) inside a solenoid, which is given by B = μ₀ * n * I, where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. Note that the direction of the magnetic field is determined by the right-hand rule and the direction of the current.
3. Step 3: Apply the formula to both solenoids. For the inner solenoid, the magnetic field inside it (r < rₐ) is Bₐ = μ₀ * nₐ * I. For the space between the solenoids (rₐ < r < r₆), the net magnetic field is the difference between the fields due to each solenoid, given by B₆ - Bₐ = μ₀ * n₆ * I - μ₀ * nₐ * I.
4. Step 4: Set up the condition given in the problem: the magnetic field inside the inner solenoid is to be in the opposite direction and have half the magnitude of the field between the solenoids. This translates to Bₐ = -1/2 * (B₆ - Bₐ).
5. Step 5: Solve the equation from Step 4 for the ratio n₆/nₐ. Substitute the expressions for Bₐ and B₆ - Bₐ, simplify the equation, and solve for n₆/nₐ.