How does the resistivity of a metal wire change if either the number of electrons per unit volume increases or the mean free time increases?

Options

A. In both cases, the resistivity will increase
B. In both cases, the resistivity will decrease
C. Increasing the number of electrons will increase the resistivity, but it will decrease iſ the mean free time
D. Increasing the number of electrons will decrease the resistivity, but it will increase if the mean free time increases

Answer

To solve the question of how the resistivity of a metal wire changes with an increase in either the number of electrons per unit volume or the mean free time, we can follow these steps:

### Step 1: Understand the formula for resistivity
The resistivity (ρ) of a metal wire is given by the formula:
\[ \rho = \frac{m}{n e^2 \tau} \]
where:
- \(m\) is the mass of the charge carriers (electrons),
- \(n\) is the number density of charge carriers (number of electrons per unit volume),
- \(e\) is the charge of the electron,
- \(\tau\) is the mean free time (relaxation time).

### Step 2: Analyze the effect of increasing the number density (n)
If the number density \(n\) increases, we can observe the relationship:
\[ \rho \propto \frac{1}{n} \]
This means that if \(n\) increases, the resistivity \(ρ\) will decrease.

### Step 3: Analyze the effect of increasing the mean free time (τ)
Now, if the mean free time \(τ\) increases, we can observe the relationship:
\[ \rho \propto \tau \]
This means that if \(τ\) increases, the resistivity \(ρ\) will also increase.

### Step 4: Combine the effects
In summary:
- Increasing the number density \(n\) leads to a decrease in resistivity \(ρ\).
- Increasing the mean free time \(τ\) leads to an increase in resistivity \(ρ\).

### Step 5: Conclusion
If both \(n\) and \(τ\) are increased, the overall effect on resistivity will depend on the relative magnitudes of the changes in \(n\) and \(τ\). However, generally:
- If \(n\) increases significantly compared to the increase in \(τ\), the resistivity will decrease.
- If \(τ\) increases significantly compared to the increase in \(n\), the resistivity will increase.

Thus, the resistivity may either increase or decrease depending on which factor has a more significant effect.