The inductance `L` of a solenoid of length `l`, whose windings are made of material of density `D` and resistivity `rho`, is (the winding resistance is`R`)

Options

A. `(mu_(0))/(4 pil) (Rm)/(rho D)`
B. `(mu_(0))/(4piR) (lm)/(rho D)`
C. `(mu_(0))/(4pil) (R^(2)m)/(rhoD)`
D. `(mu_(0))/(2 pi R) (lm)/(rhoD)`

Explanation

To find the inductance \( L \) of a solenoid of length \( l \), with windings made of a material of density \( D \) and resistivity \( \rho \), we can follow these steps:

### Step-by-Step Solution:

1. **Inductance Formula**:
The inductance \( L \) of a solenoid is given by the formula:
\[ L = \mu_0 \frac{n^2 A}{l} \]
where \( n \) is the number of turns per unit length, \( A \) is the cross-sectional area, and \( l \) is the length of the solenoid.

2. **Length of Wire**:
Let \( x \) be the length of the wire used to wind the solenoid. The number of turns \( N \) can be expressed as:
\[ N = \frac{x}{2\pi r} \]
where \( r \) is the radius of the solenoid.

3. **Resistance of Wire**:
The resistance \( R \) of the wire can be calculated using:
\[ R = \frac{\rho x}{A} \]
where \( A \) is the cross-sectional area of the wire.

4. **Mass of Wire**:
The mass \( M \) of the wire can be expressed as:
\[ M = A x D \]
where \( D \) is the density of the material.

5. **Relating Resistance and Mass**:
By substituting the expressions for \( R \) and \( M \):
\[ R \cdot M = \left(\frac{\rho x}{A}\right) \cdot (A x D) = \rho x^2 D \]
This gives us a relationship between \( R \), \( M \), \( \rho \), and \( D \).

6. **Finding Length of Wire**:
From the equation \( R \cdot M = \rho x^2 D \), we can isolate \( x \):
\[ x = \sqrt{\frac{R M}{\rho D}} \]

7. **Substituting for \( n \)**:
Substitute \( x \) back into the expression for \( n \):
\[ n = \frac{x}{2\pi r} = \frac{1}{2\pi r} \sqrt{\frac{R M}{\rho D}} \]

8. **Substituting \( n \) into Inductance Formula**:
Substitute \( n \) into the inductance formula:
\[ L = \mu_0 \frac{n^2 A}{l} = \mu_0 \frac{\left(\frac{1}{2\pi r} \sqrt{\frac{R M}{\rho D}}\right)^2 A}{l} \]

9. **Final Expression for Inductance**:
After simplifying, we find:
\[ L = \mu_0 \frac{R M}{4 \pi l \rho D} \]

10. **Comparing with Options**:
The derived expression matches with option 1:
\[ L = \frac{\mu_0}{4 \pi l} \cdot \frac{R M}{\rho D} \]

### Final Answer:
The inductance \( L \) of the solenoid is given by:
\[ L = \frac{\mu_0}{4 \pi l} \cdot \frac{R M}{\rho D} \]