Question

The length of solenoid is \( \ell \) whose windings are made of material of density \( D \) and resistivity \( Q \). The winding resistance is \( R \). The inductance of solenoid is

• A. \( \frac{\mu_0}{2 \pi \ell} \left( \frac{Rm}{Q D} \right) \)
• B. \( \frac{\mu_0}{4 \pi \ell} \left( \frac{Rm}{Q D} \right) \)
• C. \( \frac{\mu_0}{2 \pi \ell} \left( \frac{Q D}{Rm} \right) \)
• D. \( \frac{\mu_0}{4 \pi \ell} \left( \frac{Q D}{Rm} \right) \)

Hint:

The inductance of a solenoid depends on the number of turns, the area, and the length, as well as the properties of the wire used to form the solenoid.

Answer 1

The Correct Option is B

Step 1: Inductance formula for solenoid.
The inductance of a solenoid is given by the formula: \[ L = \frac{\mu_0 N^2 A}{\ell} \] where \( \mu_0 \) is the permeability of free space, \( N \) is the number of turns, \( A \) is the cross-sectional area, and \( \ell \) is the length of the solenoid.
Step 2: Relating to resistance.
The number of turns \( N \) can be related to the winding resistance \( R \), density \( D \), and resistivity \( Q \). The resulting expression for the inductance becomes: \[ L = \frac{\mu_0}{4 \pi \ell} \left( \frac{Rm}{Q D} \right) \]
Step 3: Conclusion.
The inductance of the solenoid is \( \frac{\mu_0}{4 \pi \ell} \left( \frac{Rm}{Q D} \right) \), so the correct answer is (B).