Calculating the Cross-Sectional Area of Copper Wire: A Simple Approach

When you're gearing up for your Electronics Engineering (ELEX) Board Exam, it’s not just about memorizing formulas and graphs—it's about understanding how and why they work. So, let’s talk about a fundamental concept that often trips students up: calculating the cross-sectional area of a copper wire.

You might be wondering why this matters. Well, knowing how to calculate the cross-sectional area isn’t just an academic exercise; it plays a crucial role in electrical engineering. Whether you're wiring up a circuit or designing a PCB, understanding the relationship between resistance, resistivity, and geometry is vital. In short, it’s one of those skills that may seem dry at first—but it can really light up your engineering toolbox!

Now, let’s break it down. The resistance ( R ) of a wire relates to its material properties and dimensions through the formula:

[ R = \frac{\rho \cdot L}{A} ]

This might look a bit intimidating at first, but don’t let the Greek letters fool you! Here’s what it means:

• ( R ) = Resistance in ohms (Ω)
• ( \rho ) = Resistivity of the material (in Ω-meters)
• ( L ) = Length of the wire (in meters)
• ( A ) = Cross-sectional area of the wire (in square meters)

In our example, we have a copper wire that’s 40 meters long, with a resistance of 0.25 ohms, and the resistivity of copper is given as ( 0.02 \times 10^{-6} , \Omega-m ). So how do we find the cross-sectional area ( A )?

Let’s rearrange that formula to isolate ( A ):

[ A = \frac{\rho \cdot L}{R} ]

This equation is like a recipe—mix the right amounts of given quantities to arrive at your desired output. Substituting our values in gives us:

[ A = \frac{0.02 \times 10^{-6} \Omega-m \cdot 40 m}{0.25 \Omega} ]

You can break this down step by step. First, calculate the numerator:

1. Calculate ( \rho \cdot L ): [ 0.02 \times 10^{-6} \cdot 40 = 0.0000008 , \Omega-m^2 ]

Now, we need to just divide this by the resistance:

2. Calculate ( A ): [ A = \frac{0.0000008 , \Omega-m^{2}}{0.25 , \Omega} = 0.0000032 , m^{2} ]

But hold on—this gives us the area in square meters. To convert it to square millimeters (you know, the units we often actually use), multiply by ( 1,000,000 ) (since there are ( 1,000,000 , mm^{2} ) in ( m^{2} )):

[ A = 0.0000032 , m^{2} \times 1,000,000 = 3.2 , mm^{2} ]

And there we have it! The cross-sectional area of the wire is 3.2 mm². Among the options given, this is the right answer.

Understanding these calculations is incredibly important as you prepare for your ELEX Board Exam. It’s about grasping the underlying principles of electronics so you can apply them in real-world situations. And you might even discover a deeper appreciation for how our world is wired—from the tiniest circuit board in our phones to the expansive electrical systems powering entire cities!

So, as you practice with problems like these, remember that every question is a stepping stone to mastering concepts that will be crucial in your engineering career. Embrace the challenge, and soon enough, electro-physics will feel second nature!