Question

Consider a long solenoid of length \(l\) and radius \(r\). If \(n\) is the number of turns per unit length and \(\mu_0\) is the permeability of free space, the inductance of the solenoid is:

• A. \(2\mu_0\pi n^2 r^2 l\)
• B. \(\mu_0\pi n^2 r^2 l\)
• C. \(\mu_0 n^2 r^2 l\)
• D. \(\left(\frac{\mu_0}{2\pi}\right)n^2 r^2 l\)

Hint:

Memorize inductance formula of a long solenoid. Area is \(\pi r^2\). Inductance increases with square of turns density. Larger area gives larger inductance.

Answer 1

The Correct Option is B

Concept:

• Inductance of a long solenoid is \[ L=\mu_0 n^2Al \]

• Cross-sectional area is \[ A=\pi r^2 \]

Step 1: Write standard inductance formula
\[ L=\mu_0 n^2Al \]

Step 2: Substitute area of solenoid
\[ A=\pi r^2 \] \[ L=\mu_0 n^2(\pi r^2)l \]

Step 3: Obtain final expression
\[ L=\mu_0\pi n^2r^2l \]