Animated Solution for Physics - Current Electricity: A current of $10\text{ A}$ exists in a wire of cross sectional area of $5{\text{ mm}}^2$ with a drift velocity of $2\times 10^{-3}{\text{ ms}}^{-1}$. The number of free electrons in each cubic metre of the wire is

Imagine you are looking inside a microscopic section of a conducting wire. We are given a macroscopic observable: a current of $I=10\text{ A}$ flowing through the wire. We also know the physical dimensions, specifically the cross-sectional area $A=5{\text{ mm}}^2$.

Before we proceed, we must ensure all our units are in the standard SI format to avoid any catastrophic calculation errors. Converting the area from square millimeters to square meters is a crucial first step:

Inside this wire, a sea of free electrons is constantly colliding and moving. Despite their chaotic thermal motion, the applied electric field gives them a net directional velocity, known as the drift velocity. We are given this drift velocity as $v_d=2\times 10^{-3}{\text{ ms}}^{-1}$.

To bridge the gap between the macroscopic current and the microscopic electron motion, we rely on a fundamental relationship in current electricity. The current $I$ is directly proportional to the number of free electrons per unit volume $n$, the elementary charge $e$, the cross-sectional area $A$, and the drift velocity $v_d$. This gives us our master equation:

Our objective is to find $n$, the electron density. We know the elementary charge of an electron is a universal constant, $e=1.6\times 10^{-19}\text{ C}$.

Now, let's carefully substitute our known values into the equation.

We need to isolate $n$. Bringing all the numerical terms to the denominator on the left side, we get:

This might look like a heavy calculation, but let's break it down by grouping the powers of 10 and the standard numbers. Notice how $5\times 2=10$. This perfectly cancels out the $10$ in the numerator!

To match the format of our multiple-choice options, we adjust the decimal point by shifting it three places to the right, which decreases the exponent by three:

This massive number represents the sheer density of free electrons available for conduction in a typical metal. Understanding this microscopic perspective is key to mastering current electricity!