A conducting wire of length L, uniform area of cross-section A of a material having n free electrons per unit volume offers a resistance R to flow of current through itself. (m and e respectively denote the mass and charge of electron). If \( \tau \) is mean free time of free electrons in the conductor, the correct formula for resistance R is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( R = \displaystyle\frac{mL}{e^{2}nA\tau} \)
In the drift model of conduction, an electron accelerates in the field for the mean free time \( \tau \), so its drift velocity is \( v_d = eE\tau/m \). Putting this in \( I = neAv_d \):
\( \displaystyle I = neA\,\frac{eE\tau}{m} = \frac{ne^{2}\tau}{m}\,A\,E \)
With \( E = V/L \) for a wire of length \( L \):
\( \displaystyle I = \frac{ne^{2}\tau A}{mL}\,V \;\Rightarrow\; R = \frac{V}{I} = \frac{mL}{ne^{2}A\tau} \)
This is \( R = \rho L/A \) with resistivity \( \rho = m/(ne^{2}\tau) \).
Shortcut: Dimensional check on the geometry: \( R \) must be proportional to \( L \) and inversely proportional to \( A \). Only Option (1) has \( L \) in the numerator and \( A \) in the denominator.