Question

You are required to design an air-filled solenoid of inductance \( 0.016 \, \text{H} \) having a length \( 0.81 \, \text{m} \) and radius \( 0.02 \, \text{m} \). The number of turns in the solenoid should be:

• A. 2592
• B. 2866
• C. 2976
• D. 3140

Hint:

The inductance of a solenoid depends on the number of turns, the cross-sectional area, and the length of the solenoid. Use the formula \( L = \mu_0 \frac{N^2 A}{l} \) to solve for the unknowns.

Answer 1

The Correct Option is B

The inductance \( L \) of a solenoid is given by the formula: \[ L = \mu_0 \frac{N^2 A}{l} \] where: - \( L \) is the inductance, - \( \mu_0 \) is the permeability of free space (\( 4 \pi \times 10^{-7} \, \text{T m/A} \)), - \( N \) is the number of turns, - \( A \) is the cross-sectional area of the solenoid, - \( l \) is the length of the solenoid. The area \( A \) of the solenoid with radius \( r \) is: \[ A = \pi r^2 = \pi (0.02)^2 = 1.256 \times 10^{-3} \, \text{m}^2 \] We are given \( L = 0.016 \, \text{H} \) and \( l = 0.81 \, \text{m} \). Substituting the known values into the formula: \[ 0.016 = (4 \pi \times 10^{-7}) \frac{N^2 (1.256 \times 10^{-3})}{0.81} \] Solving for \( N^2 \), we get: \[ N^2 = \frac{0.016 \times 0.81}{(4 \pi \times 10^{-7}) \times (1.256 \times 10^{-3})} = 2866 \] Therefore, the number of turns \( N \) is approximately 2866. Thus, the correct answer is (B).