Animated Solution for Physics - Current Electricity: In a conductor, if the number of conduction electrons per unit volume is $8.5\times 10^{28}{\text{ m}}^{-3}$ and mean free time is $25\text{ fs}$ (femto second), it's approximate resistivity is (Take, $m_e=9.1\times 10^{-31}\text{ kg}$)

Imagine the view inside a metallic conductor. It is not a quiet, empty highway. Instead, it is a bustling, chaotic environment where free electrons are constantly zipping around, colliding with the massive, vibrating positive lattice ions.

These collisions are the fundamental cause of electrical resistance. The average time an electron gets to travel freely between two successive collisions is a crucial parameter known as the mean free time, denoted by $\tau$.

Bridging the Gap

To connect this microscopic chaos to the macroscopic property of resistivity ($\rho$), we rely on the elegant Drude model. The model gives us a powerful master equation:

Here, $m_e$ is the mass of an electron, $n$ is the number density of free electrons, $e$ is the elementary charge, and $\tau$ is the mean free time. Notice how resistivity is inversely proportional to both the electron density and the mean free time. More electrons or longer times between collisions mean less resistance to the flow of current.

Let's carefully substitute the given values into our master equation. We are given $n=8.5\times 10^{28}{\text{ m}}^{-3}$, $\tau =25\text{ fs}=25\times 10^{-15}\text{ s}$, $m_e=9.1\times 10^{-31}\text{ kg}$, and $e=1.6\times 10^{-19}\text{ C}$.

This is where we must be vigilant to avoid silly mistakes, especially with the powers of 10. First, let's square the charge of the electron:

Now, let's group the numerical coefficients and the powers of 10 in the denominator separately:

Calculating the numerical part: $8.5\times 2.56\times 25=544$. Calculating the exponent: $28-38-15=-25$.

So, the denominator simplifies beautifully to $544\times 10^{-25}$.

Now, we bring the numerator back into the picture:

To match the standard scientific notation of our options, we shift the decimal point two places to the right, which decreases the exponent by 2:

Looking at the given options, the closest order of magnitude is clearly $10^{-8}\Omega \text{-m}$. This confirms that option (d) is the correct answer.

As a thought experiment, consider what happens when you heat the conductor. The lattice ions vibrate more vigorously, causing the electrons to collide more frequently. This decreases $\tau$, which in turn increases the resistivity $\rho$. This microscopic view perfectly explains why the resistance of metals increases with temperature!