Question

If current ' i ' is passing through the solenoid of diameter ' \(d\) ' having number of turns per unit length ' \(n\) ', then the inductance per unit length near the middle of a long solenoid is directly proportional to

• A. d and n
• B. \(\mathrm{d}^{2}\) and n
• C. d and \(\mathrm{n}^{2}\)
• D. \(\mathrm{d}^{2}\) and \(\mathrm{n}^{2}\)

Hint:

Self inductance of a solenoid is \(\mu_0 n^2 A l\), so per unit length it is \(\mu_0 n^2 A\) with \(A\propto d^2\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
For a long solenoid of length \(l\), \(L=\mu_0 n^2 A\,l\).

Step 2: Per unit length
\(\dfrac{L}{l}=\mu_0 n^2 A\).

Step 3: Area
Cross-section is a circle of diameter \(d\): \(A=\dfrac{\pi d^2}{4}\).

Step 4: Dependence
\(\dfrac{L}{l}=\dfrac{\mu_0\pi}{4}\,n^2d^2\), so it is proportional to \(d^2\) and \(n^2\).

Final Answer:
The inductance per unit length is proportional to \(d^2\) and \(n^2\). \[ \boxed{\text{(D) }\text{d}^{2}\text{ and n}^{2}} \]