Mobility of free electrons in a current carrying conductor is proportional to
(1) Relaxation time (2) Electric field
(3) Potential difference (4) All of these

Explanation

To solve the question regarding the mobility of free electrons in a current-carrying conductor, let's break it down step by step.

### Step 1: Understand the Definition of Mobility
Mobility (μ) of free electrons is defined as the drift velocity (Vd) per unit electric field (E). Mathematically, this is expressed as:
\[ \mu = \frac{V_d}{E} \]

**Hint:** Recall the relationship between drift velocity and electric field.

### Step 2: Express Drift Velocity
The drift velocity (Vd) can be expressed in terms of the charge of the electron (e), electric field (E), mass of the electron (m), and relaxation time (τ):
\[ V_d = \frac{eE\tau}{m} \]

**Hint:** Remember that drift velocity is influenced by the electric field and relaxation time.

### Step 3: Substitute Drift Velocity into Mobility Equation
Now, substitute the expression for drift velocity back into the mobility equation:
\[ \mu = \frac{V_d}{E} = \frac{eE\tau/m}{E} \]
When we simplify this, the electric field (E) cancels out:
\[ \mu = \frac{e\tau}{m} \]

**Hint:** Look for terms that can be simplified in the equation.

### Step 4: Analyze the Result
From the equation \( \mu = \frac{e\tau}{m} \), we can see that mobility (μ) is directly proportional to the relaxation time (τ). This indicates that the mobility of free electrons in a conductor primarily depends on the relaxation time and is independent of the electric field and potential difference.

**Hint:** Identify the key variable that affects mobility based on the derived equation.

### Step 5: Conclusion
Based on the analysis, the mobility of free electrons in a current-carrying conductor is proportional to relaxation time (τ). Therefore, the correct answer to the question is:
**(1) Relaxation time.**

**Final Answer:** The mobility of free electrons in a current-carrying conductor is proportional to relaxation time (Option 1).