Question

The self-inductance of a long solenoid of cross-sectional area \( A \), length \( l \) and \( n \) turns per unit length is given by:

• A. \( \mu_0 n A l \)
• B. \( \mu_0 n^2 A l \)
• C. \( \mu_0 n^2 A^2 l \)
• D. \( \mu_0 n^2 \pi A^2 l \)

Hint:

When dealing with solenoids, remember that the self-inductance depends on the number of turns per unit length, the cross-sectional area, and the length of the solenoid. The key formula is \( L = \mu_0 n^2 A l \), where \( n \) is the turns per unit length.

Answer 1

The Correct Option is B

Step 1: Understanding the formula for self-inductance The self-inductance \( L \) of a solenoid is given by the formula: \[ L = \frac{\mu_0 N^2 A}{l} \] where:
\( L \) is the self-inductance,
\( \mu_0 \) is the permeability of free space,
\( N \) is the total number of turns in the solenoid,
\( A \) is the cross-sectional area of the solenoid,
\( l \) is the length of the solenoid.

Step 2: Relating the number of turns \( N \) to the turns per unit length \( n \)
The total number of turns \( N \) is related to the turns per unit length \( n \) and the length of the solenoid \( l \) by the equation: \[ N = n \cdot l \] where \( n \) is the number of turns per unit length, and \( l \) is the length of the solenoid.
Step 3: Substitute \( N = n \cdot l \) into the formula for \( L \)
Now, substitute \( N = n \cdot l \) into the formula for \( L \): \[ L = \frac{\mu_0 (n \cdot l)^2 A}{l} \] Simplifying the expression: \[ L = \mu_0 n^2 A l \] Thus, the self-inductance of the solenoid is \( \mu_0 n^2 A l \).