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: Recall the formula for electrical resistivity.
From the classical free electron theory, resistivity is given by: \[ \rho = \frac{m}{n e^2 \tau} \] where \( m \) is mass of electron, \( e \) is charge of electron, \( n \) is number of free electrons per unit volume, and \( \tau \) is relaxation time.

Step 2: Identify proportionality.
From the formula: \[ \rho \propto \frac{1}{n\tau} \] So resistivity is inversely proportional to both \( n \) and \( \tau \).

Step 3: Analyze each given parameter.
- \( n \): appears in denominator → \( \rho \propto \frac{1}{n} \)
- \( \tau \): also in denominator → \( \rho \propto \frac{1}{\tau} \)
- \( A \): does not appear in resistivity expression
- \( T \): affects \( \tau \) indirectly but not directly in formula

Step 4: Eliminate incorrect options.
Resistivity is not directly proportional to \( n \), \( \tau \), or \( A \).
It is inversely proportional to \( n \) and \( \tau \).

Step 5: Choose the correct direct proportionality form.
Since \[ \rho \propto \frac{1}{n} \] this matches one of the given options.

Step 6: Physical interpretation.
More free electrons (\( n \uparrow \)) means easier current flow → lower resistivity.
Thus resistivity decreases as \( n \) increases.

Step 7: Final conclusion.
Hence, \[ \boxed{\rho \propto \frac{1}{n}} \] Therefore, the correct option is \[ \boxed{(4)\ \dfrac{1}{n}} \]