Problem 32 - Technical Mathematics with Calculus

Question

Applications. The resistance \(R\) of a conductor is a function of temperature: $$ f(t)=R_{0}(1+\alpha t) $$ where \(R_{0}\) is the resistance at \(0^{\circ} \mathrm{C}\) and \(\alpha\) is the temperature coefficient of resistance \(\left(0.00427 \text { for copper). If the resistance of a copper coil is } 9800 \Omega \text { at } 0^{\circ} \mathrm{C}\right.\) find \(f(20.0), f(25.0),\) and \(f(30.0)\)

Step-by-Step Solution

Answer

\(f(20.0)=10164 \Omega\), \(f(25.0)=10245.25 \Omega\), and \(f(30.0)=10326.8 \Omega\).

Step 1: Identify the known values

The known values from the problem are the resistance at 0 degrees Celsius, which is \(R_0=9800 \, \Omega\), and the temperature coefficient of resistance for copper, \(\alpha=0.00427\).

Step 2: Plug in the values for \(f(20.0)\)

Use the formula \(f(t)=R_{0}(1+\alpha t)\) to find the resistance at 20.0 degrees Celsius. Substitute \(t=20.0\), \(R_0=9800 \, \Omega\), and \(\alpha=0.00427\) into the formula to calculate \(f(20.0)\).

Step 3: Calculate \(f(20.0)\)

Perform the calculation: \[f(20.0) = 9800 (1 + 0.00427 \times 20.0)\]. Find the product of \(0.00427\) and \(20.0\) and then multiply the sum of this product and 1 by \(9800 \, \Omega\).

Step 4: Plug in the values for \(f(25.0)\)

Similarly, substitute \(t=25.0\) into the formula to calculate \(f(25.0)\).

Step 5: Calculate \(f(25.0)\)

Perform the calculation: \[f(25.0) = 9800 (1 + 0.00427 \times 25.0)\].

Step 6: Plug in the values for \(f(30.0)\)

Finally, substitute \(t=30.0\) into the formula to calculate \(f(30.0)\).

Step 7: Calculate \(f(30.0)\)

Perform the calculation: \[f(30.0) = 9800 (1 + 0.00427 \times 30.0)\].

Key Concepts

Understanding Electrical Resistance

Electrical resistance is a fundamental concept in the field of electronics, crucial for understanding how current flows through different materials. Think of resistance as a type of friction for electricity. Just as friction makes it tougher for objects to move across surfaces, resistance makes it harder for electrons to flow through a conductor.

Ohm's law, which states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance, forms the cornerstone of electrical resistance understanding.

In mathematical terms, it is represented as \( V = IR \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. Higher resistance means less current flow for a given voltage. When we talk about a copper coil's resistance at \(0^{\text{c}}\) being \(9800\text{Ω}\), we're saying that at this temperature, the coil will strongly oppose the flow of electrical current.

Conductivity and Temperature

Conductivity and temperature are inherently linked in materials like metals. As temperature increases, metals tend to have higher energy atoms that vibrate more. This vibration can scatter the electrons more, which usually increases the resistance of the material.

Each material has its own temperature coefficient of resistance, denoted as \(\text{α}\), which quantifies how much its resistance changes with temperature. For example, copper has a high conductivity but also a noticeable change in resistance with temperature, with a coefficient of \(0.00427\text{/}^{\text{c}}\). This means that for each degree Celsius increase in temperature, the resistance of copper will increase by a factor of 0.00427.

Why Temperature Matters

Understanding the temperature dependence is important in designing circuits that operate efficiently under different conditions. Imagine a wire that operates perfectly at room temperature but becomes excessively resistive and less efficient on a hot day - knowing about the temperature coefficient can help engineers preemptively address such challenges.

Resistance Calculation

Calculations involving resistance are central to electrical engineering and physics. When you're presented with a problem like finding the resistance of a conductor at different temperatures, you start by using the fundamental formula discussed earlier: \(f(t)=R_{0}(1+\alpha t)\), where \(f(t)\) is the resistance at temperature \(t\), \(R_{0}\) is the initial resistance, and \(\alpha\) is the temperature coefficient of resistance.

Breaking Down the Steps

To solve for any temperature, you simply plug the corresponding value of \(t\) into the equation and calculate, as seen in the exercise solution steps above. The key is understanding how the initial resistance \(R_{0}\) and the temperature coefficient \(\alpha\) relate to the overall resistance change.

For instance, the solution involved calculating the resistance of a copper coil at several temperatures. It's not just about plugging in numbers but understanding the exponential growth of resistance with temperature, exemplified by the given formula. By utilizing the temperature coefficient correctly, one can accurately determine how a conductor's resistance will alter under various thermal conditions.