Two air-cored solenoids are designed to have the exact same self-inductance. The first solenoid has one-fourth the length of the second solenoid, while their cross-sectional areas are identical. The ratio of the total number of turns in the first solenoid to that in the second solenoid is

Step-by-step solution

The self-inductance of a solenoid in terms of total number of turns N is given by L = ₀ N^2 A l . Since both solenoids are air-cored and have the same self-inductance ( L₁ = L₂ ) and identical cross-sectional areas ( A₁ = A₂ ), we have: N₁^2 l₁ = N₂^2 l₂ ( N₁ N₂ )^2 = l₁ l₂ Given that l₁ = l₂ 4 , we get l₁ l₂ = 1 4 . N₁ N₂ = 1 4 = 1 2 Thus, the ratio of the total number of turns is 1:2 . Answer: 1:2