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, we will analyze the relationship between mobility and the factors mentioned in the options.

### Step-by-Step Solution:

1. **Understanding Mobility**:
Mobility (μ) of electrons in a conductor is defined as the ability of electrons to move through the 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 and Electric Field**:
The drift velocity \( V_d \) of electrons is related to the electric field \( E \) and the relaxation time \( \tau \) (the average time between collisions of electrons) by the equation:
\[ V_d = \frac{eE\tau}{m} \]
where \( e \) is the charge of the electron and \( m \) is its mass. This shows that \( V_d \) is directly proportional to both \( E \) and \( \tau \).

3. **Proportionality to Relaxation Time**:
The relaxation time \( \tau \) is influenced by temperature; as temperature increases, the relaxation time decreases due to more frequent collisions. Thus, mobility is also directly proportional to the relaxation time.

4. **Conclusion**:
From the above relationships, we can conclude that the mobility of free electrons in a current-carrying conductor is proportional to:
- Relaxation time \( \tau \)
- Electric field \( E \)

Since the mobility is influenced by both the relaxation time and the electric field, the correct answer is:
**(4) All of these**.