Consider the follwing three statmenet for a current carrying conductor of non-uniform cross-section.
STATEMENT-1 `:` Current density at any cross -section of the conductor is proportional to electric field at the cross-section.
STATEMENT-2 `:` If temperature of the conductor is increased, relaxation time of drifting electron decreases.
STATEMENT-3 `:` Electrons in the conductor flow from low potential to high potential.

Options

A. F F F
B. F T F
C. F F T
D. T T T

Answer

To analyze the three statements regarding a current-carrying conductor of non-uniform cross-section, we will evaluate each statement step by step.

### Step 1: Evaluate Statement 1
**Statement 1:** Current density at any cross-section of the conductor is proportional to the electric field at the cross-section.

**Solution:**
- The current density \( J \) is defined as the current \( I \) per unit area \( A \), given by the formula:
\[ J = \frac{I}{A} \]
- For a conductor, we can express the current \( I \) in terms of charge density \( n \), charge \( e \), and drift velocity \( V_d \):
\[ I = n \cdot e \cdot A \cdot V_d \]
- Substituting this into the equation for current density:
\[ J = \frac{n \cdot e \cdot A \cdot V_d}{A} = n \cdot e \cdot V_d \]
- The drift velocity \( V_d \) can be expressed in terms of the electric field \( E \) and relaxation time \( \tau \):
\[ V_d = \frac{eE}{m} \cdot \tau \]
- Substituting \( V_d \) back into the equation for \( J \):
\[ J = n \cdot e \cdot \left(\frac{eE}{m} \cdot \tau\right) = \frac{n \cdot e^2 \cdot \tau}{m} \cdot E \]
- This shows that \( J \) is directly proportional to \( E \). Therefore, **Statement 1 is correct**.

### Step 2: Evaluate Statement 2
**Statement 2:** If the temperature of the conductor is increased, the relaxation time of drifting electrons decreases.

**Solution:**
- The resistivity \( \rho \) of a conductor is given by:
\[ \rho = \frac{m}{n e^2 \tau} \]
- As temperature increases, the resistivity \( \rho \) generally increases for conductors. This implies:
\[ \rho \propto 1/\tau \implies \tau \propto \frac{1}{\rho} \]
- Therefore, if \( \rho \) increases, \( \tau \) must decrease. Thus, **Statement 2 is correct**.

### Step 3: Evaluate Statement 3
**Statement 3:** Electrons in the conductor flow from low potential to high potential.

**Solution:**
- The conventional current \( I \) flows from high potential to low potential. However, electrons, being negatively charged, move in the opposite direction to the current.
- Hence, while current flows from high potential to low potential, electrons flow from low potential to high potential. Therefore, **Statement 3 is correct**.

### Conclusion
All three statements are correct.