If `mu ` is the mobility of free electrons inside a conductor , then `mu` is independent of

Options

A. Area of cross-section of conductor
B. Length of conductor
C. Volume of conductor
D. All of these

Answer

To determine what the mobility (μ) of free electrons inside a conductor is independent of, we can analyze the relationship between mobility and various physical parameters.

### Step-by-Step Solution:

1. **Understanding Mobility**:
- Mobility (μ) is defined as the ability of free electrons to move through a conductor when an electric field is applied. It is mathematically expressed as:
\[ \mu = \frac{v_d}{E} \]
where \( v_d \) is the drift velocity of the electrons and \( E \) is the electric field.

2. **Drift Velocity**:
- The drift velocity \( v_d \) can be expressed in terms of current (I), area (A), electron density (n), and charge of an electron (e):
\[ v_d = \frac{I}{n \cdot A \cdot e} \]

3. **Electric Field**:
- The electric field \( E \) can be expressed as:
\[ E = \frac{V}{L} \]
where \( V \) is the potential difference and \( L \) is the length of the conductor.

4. **Substituting into Mobility Formula**:
- Substituting the expressions for \( v_d \) and \( E \) into the mobility formula gives:
\[ \mu = \frac{I}{n \cdot A \cdot e} \cdot \frac{L}{V} \]

5. **Rearranging the Formula**:
- Rearranging the formula, we have:
\[ \mu = \frac{I \cdot L}{n \cdot A \cdot e \cdot V} \]

6. **Identifying Dependencies**:
- From the rearranged formula, we can see that mobility (μ) depends on:
- Electron density (n)
- Charge of the electron (e)
- Current (I)
- Potential difference (V)
- However, it does not depend on the length (L) or area (A) of the conductor.

7. **Conclusion**:
- Therefore, mobility (μ) is independent of the length and area of the conductor, as well as the volume of the conductor.

### Final Answer:
- Mobility (μ) is independent of:
- Length of the conductor
- Area of the conductor
- Volume of the conductor
- Thus, the correct answer is **option D: All of these**.