Question

The resistivity of a metallic wire is directly proportional to (T – temperature; \( \tau \) average time of collisions of free electrons; \( n \) – number of free electrons per unit volume; \( A \) – area of cross-section)

• A. \( n \)
• B. \( \tau \)
• C. \( A \)
• D. \( \dfrac{1}{n} \)
• E. \( \dfrac{1}{T} \)

Hint:

Remember: \[ \rho = \frac{m}{ne^2\tau} \] Resistivity decreases if number of free electrons or relaxation time increases.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Resistivity (\( \rho \)) is an intrinsic property of a material that quantifies how strongly it resists the flow of electric current. It is related to the microscopic properties of the material, such as the density of charge carriers and the frequency of their collisions.
Step 2: Key Formula or Approach:
The formula for resistivity derived from the Drude model of electrical conduction is:
\[ \rho = \frac{m}{ne^2\tau} \] where:
- \( m \) is the mass of an electron (constant).
- \( n \) is the number of free electrons per unit volume (charge carrier density).
- \( e \) is the elementary charge (constant).
- \( \tau \) is the average time between collisions (relaxation time).
Step 3: Detailed Explanation:
From the formula \( \rho = \frac{m}{ne^2\tau} \), we can analyze the proportionalities:
- Resistivity \( \rho \) is inversely proportional to the number density of free electrons, n. \( \rho \propto \frac{1}{n} \). This corresponds to option (D).
- Resistivity \( \rho \) is inversely proportional to the average collision time, \( \tau \). \( \rho \propto \frac{1}{\tau} \). This corresponds to option (E).
The question asks what resistivity is "directly proportional to". This is ambiguous as the options include \( \frac{1}{n} \) and \( \frac{1}{\tau} \). However, for a metallic wire, as the temperature (T) increases, the thermal agitation of the atoms increases. This causes the free electrons to collide more frequently with the lattice ions, which means the average time between collisions, \( \tau \), decreases. Since \( \tau \) decreases with increasing T, and \( \rho \propto \frac{1}{\tau} \), the resistivity \( \rho \) increases with temperature T. This is a well-known property of metals. Therefore, the dependence on temperature is directly linked to the dependence on \( \frac{1}{\tau} \). Option (E) is the most direct physical relationship described. For a given metal, n is essentially constant, whereas \( \tau \) is strongly dependent on temperature. The area A affects the resistance (R = \( \rho \frac{L}{A} \)), not the resistivity (\(\rho\)).
Between options (D) and (E), the temperature dependence of resistivity in metals is primarily explained by the change in \( \tau \), making \( 1/\tau \) the more significant factor in discussions about what resistivity is proportional to.
Step 4: Final Answer:
Resistivity is directly proportional to \( \frac{1}{\tau} \) (and also to \( \frac{1}{n} \)). Given the options, and the physical reason for temperature dependence, \( \frac{1}{\tau} \) is a very strong candidate. Let's select E based on this reasoning.