The resistance of a metallic conductor increases with temperature due to.

Options

A. Electron density increases
B. Relaxation time decreases
C. Length of wire decreases
D. All of these

Answer

To solve the question regarding why the resistance of a metallic conductor increases with temperature, we can break down the explanation step by step:

### Step-by-Step Solution:

1. **Understanding Resistance and Current**:
- Resistance (R) is a measure of how much a material opposes the flow of electric current (I). The relationship between current and resistance is given by Ohm's Law: \( V = IR \), where V is the voltage across the conductor.

**Hint**: Remember that higher resistance means less current for a given voltage.

2. **Role of Electrons in Conductors**:
- In a metallic conductor, electric current is carried by free electrons. The number of electrons that can flow through a cross-sectional area of the conductor per unit time determines the current.

**Hint**: Think of electrons as the charge carriers in a conductor.

3. **Drift Velocity and Relaxation Time**:
- The drift velocity (v_d) of electrons is the average velocity of electrons due to an electric field. It is influenced by the relaxation time (τ), which is the average time between collisions of electrons with atoms in the conductor.
- The formula for drift velocity is:
\[ v_d = \frac{eE\tau}{m} \]
where \( e \) is the charge of the electron, \( E \) is the electric field, and \( m \) is the mass of the electron.

**Hint**: Higher drift velocity means more current, and relaxation time is crucial for determining how quickly electrons can move through the conductor.

4. **Effect of Temperature on Collisions**:
- As temperature increases, the thermal energy of the electrons also increases. This leads to more frequent collisions between electrons and the lattice atoms of the conductor.
- Consequently, the relaxation time (τ) decreases because the time between collisions becomes shorter.

**Hint**: Higher temperature means more energy and more collisions, which affects how long electrons can travel without hitting something.

5. **Relationship Between Resistance and Relaxation Time**:
- Since current (I) is proportional to drift velocity (v_d), and resistance (R) is inversely proportional to current, we can establish that:
\[ R \propto \frac{1}{\tau} \]
- Therefore, if relaxation time decreases (due to increased collisions at higher temperatures), the resistance must increase.

**Hint**: Remember that if τ decreases, R must increase.

6. **Conclusion**:
- The increase in resistance of a metallic conductor with temperature is primarily due to the decrease in relaxation time (τ) caused by increased collisions among electrons.

**Final Answer**: The resistance of a metallic conductor increases with temperature due to the decrease in relaxation time.