Problem 52 - Basic Engineering Mathematics

Question

The electrical resistance \(R\) of a piece of wire is inversely proportional to the cross-sectional area \(A\). When \(A=5 \mathrm{~mm}^{2}, R=7.02\) ohms. Determine (a) the coefficient of proportionality and (b) the cross-sectional area when the resistance is \(4 \mathrm{ohms}\).

Step-by-Step Solution

Answer

(a) The coefficient is 35.1. (b) The area is 8.775 mm².

Step 1: Understanding Proportionality

Since resistance \(R\) is inversely proportional to the cross-sectional area \(A\), we can express this relationship as \(R = \frac{k}{A}\), where \(k\) is the coefficient of proportionality.

Step 2: Finding the Coefficient of Proportionality

Given that \(R = 7.02\) ohms when \(A = 5 \mathrm{~mm}^2\), substitute these values into the equation: \[ 7.02 = \frac{k}{5} \] Multiply both sides by 5 to solve for \(k\): \[ k = 7.02 \times 5 = 35.1 \] Thus, the coefficient of proportionality \(k\) is 35.1.

Step 3: Using the Coefficient to Find New Area

Now, we need to find the cross-sectional area when the resistance is \(4\) ohms. Set the equation using the coefficient \(k\): \[ 4 = \frac{35.1}{A} \] Multiply both sides by \(A\) and then divide by 4 to isolate \(A\): \[ A = \frac{35.1}{4} \] Calculate: \[ A = 8.775 \text{ mm}^2 \] Hence, the cross-sectional area when the resistance is 4 ohms is \(8.775 \text{ mm}^2\).

Key Concepts

Electrical Resistance

Electrical resistance is a measure of how much a material opposes the flow of electric current. Imagine trying to push water through a clogged pipe. The clog represents resistance; the more blocked the pipe, the harder it is for water to flow. In a similar way, resistance affects how easily electricity can travel through a wire.

• Different materials have different levels of resistance.
• Metals usually have low resistance, making them good conductors of electricity.
• Resistance is measured in ohms (Ω).

Understanding resistance is crucial for designing electrical circuits. It helps in determining how much energy will be used and how much heat will be generated within a circuit.
Resistance changes with material, size, temperature, and impurities. But a key relationship is how it varies with the cross-sectional area—this forms the basis of the inverse proportionality discussed in exercises like the one given above.

Cross-Sectional Area

The cross-sectional area of a wire plays a significant role in determining its electrical resistance. Think of it as the thickness of the wire. The larger the area, the less resistance there is to current flow. This is because more area allows more paths for electrons to travel through, making it easier for electricity to pass.

• A thicker wire (larger cross-sectional area) has lower resistance.
• A thinner wire (smaller cross-sectional area) has higher resistance.

In the given exercise, the cross-sectional area is measured in square millimeters (mm²). When the area was smaller (5 mm²), a resistance of 7.02 ohms was noted. However, when the resistance needed to change to 4 ohms, the area had to increase to 8.775 mm², demonstrating the inverse relationship between area and resistance.

Coefficient of Proportionality

The coefficient of proportionality serves as a constant in the formula that relates resistance and cross-sectional area. This coefficient, denoted as "k," provides a specific value by which the inverse relationship between resistance and area is quantified.

• The formula describing the relationship is: r>\[ R = \frac{k}{A} \]
• In the given exercise, the coefficient was calculated to be 35.1.
• This calculation was based on the given resistance (7.02 ohms) and known area (5 mm²).

This coefficient remains constant for a given material and conditions, allowing us to predict how changes in area affect resistance. For instance, knowing "k" allows you to "plug in" a new resistance value into the formula to find the associated cross-sectional area, as we did to calculate an area of 8.775 mm² for a resistance of 4 ohms.