Animated Solution for Physics - Current Electricity: A current of $5\text{ A}$ passes through a copper conductor (resistivity $=1.7\times 10^{-8}\Omega \text{-m}$) of radius of cross-section $5\text{ mm}$. Find the mobility of the charges, if their drift velocity is $1.1\times 10^{-3}\text{ m/s}$.

Imagine you are looking inside a copper wire. It's not just a static piece of metal; it's a bustling highway of electrons! We are given a copper conductor carrying a steady current $I=5\text{ A}$. The wire has a radius $r=5\text{ mm}$, which we must immediately convert to standard SI units as $5\times 10^{-3}\text{ m}$.

The material itself has a specific resistivity, $\rho =1.7\times 10^{-8}\Omega \text{-m}$, which tells us how strongly the copper atoms resist the flow of these electrons. Despite this resistance, the electrons manage to drift forward with a velocity $v_d=1.1\times 10^{-3}\text{ m/s}$. Our mission is to find the mobility ($\mu$) of these charge carriers.

Mobility is a beautiful concept. It tells us how fast an electron can drift for a given push. That "push" is the electric field ($E$). Mathematically, mobility is defined as:

We know the drift velocity, but the electric field $E$ is hiding. How do we find it? We call upon the microscopic form of Ohm's Law, which connects the electric field to the current density ($J$) and resistivity ($\rho$):

Since current density is simply the total current divided by the cross-sectional area ($J=\frac{I}{A}$), we can rewrite the electric field as:

Now, let's substitute this back into our mobility equation. This is where the magic happens:

Since the wire is cylindrical, its cross-sectional area is $A=\pi r^2$. Plugging this in gives us our master equation:

Now, we just need to carefully substitute our known values into the master equation. Watch out for the powers of ten!

Let's break down the numerator first. Squaring the radius gives $25\times 10^{-6}$. Multiplying this by $1.1\times 10^{-3}$ and $\pi$ (approximately $3.1415$) yields:

Now for the denominator:

Dividing the two gives us our final mobility:

Looking at our options, the closest value is $1.0{\text{ m}}^2/\text{V-s}$. The physics works out perfectly!