Question

At what temperature will the resistance of a copper wire be three times its value at \(0^{\circ} \mathrm{C}\) ? (Given: temperature coefficient of resistance for copper \(=4 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\) ) (A) \(400^{\circ} \mathrm{C}\) (B) \(450^{\circ} \mathrm{C}\) (C) \(500^{\circ} \mathrm{C}\) (D) \(550^{\circ} \mathrm{C}\)

Short answer

The temperature at which the resistance of a copper wire is three times its value at \(0^{\circ} \mathrm{C}\) is (C) \(500^{\circ} \mathrm{C}\).

Step-by-step solution

Write down the given values and resistance formula

We are given: - Temperature coefficient of resistance for copper, \(α = 4 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\) - We want to find the temperature \(t\) at which the resistance is three times its value at \(0^{\circ} \mathrm{C}\). The formula relating the resistance of a wire with its temperature coefficient and the change in temperature is: \(R_t = R_0 (1 + α(t - t_0))\) where \(R_t\) is the resistance at temperature \(t\), \(R_0\) is the resistance at temperature \(t_0\), and \(α\) is the temperature coefficient of resistance.

Set up the equation for resistance at the desired temperature

Let's denote the resistance at \(0^{\circ} \mathrm{C}\) as \(R_0\) and the resistance at the desired temperature as \(R_3\), where \(R_3\) is three times the initial resistance, i.e., \(R_3 = 3R_0\). The equation for resistance at the desired temperature is: \(R_3 = R_0 (1 + α(t - t_0))\)

Substitute the known values into the equation

Since \(R_3 = 3R_0\), we can replace \(R_3\) with \(3R_0\) and substitute the initial temperature \(t_0 = 0^{\circ} \mathrm{C}\) and the temperature coefficient \(α = 4 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\): \(3R_0 = R_0 (1 + (4 \times 10^{-3})(t - 0))\)

Solve the equation for the desired temperature \(t\)

Next, we will solve the equation for the desired temperature \(t\). We can divide both sides by \(R_0\) to simplify the equation: \(3 = 1 + 4 \times 10^{-3} t\) Now, subtract 1 from both sides of the equation: \(2 = 4 \times 10^{-3} t\) Finally, divide both sides by \(4 \times 10^{-3}\) to obtain the value of \(t\): \( t = \frac{2}{4 \times 10^{-3}} = 500\) The temperature at which the resistance of the copper wire is three times its initial value is \(t = 500^{\circ} \mathrm{C}\). So, the correct answer is (C) \(500^{\circ} \mathrm{C}\).

Key concepts

Copper wire resistance

Copper wire is often used in electrical circuits because it has low resistance and is an excellent conductor of electricity. However, like any material, copper's resistance isn't static. In fact, resistance can change with various factors, with temperature being a primary one.
The resistance of a copper wire at a given temperature depends on its physical dimensions and material properties. To understand how resistance works:

• **Length of Wire:** The longer the wire, the higher the resistance.
• **Cross-Sectional Area:** Thinner wires have higher resistance.
• **Material:** Different materials have different innate resistances. Copper is popular due to its relatively low resistivity.

The initial resistance of a copper wire at a reference temperature (often at 0°C for simplicity) is denoted as \(R_0\). Afterwards, any changes in temperature can lead to changes in this initial resistance value.

Temperature coefficient

The temperature coefficient of resistance is an important factor that defines how much a material's resistance changes with temperature. For copper, this is given as \(4 \times 10^{-3}\) \({}^{\circ}\)C\(^{-1}\).
This coefficient tells you that for every 1°C increase in temperature, copper's resistance will increase by \(4 \times 10^{-3}\) times its original resistance.
Understanding this concept allows us to predict and calculate how resistance will change with temperature changes.
**Why is the temperature coefficient crucial?**

• **Predictability:** Knowing the coefficient helps in designing circuits that may operate under varying temperatures.
• **Safety and Efficiency:** Understanding potential changes in resistance can prevent overheating and ensure efficient functioning of electrical systems.

Ohm's Law

Ohm's Law is a fundamental principle in physics and electrical engineering and expresses the relationship between voltage, current, and resistance in an electrical circuit. This law is stated as:\[ V = IR \]where:

• \(V\) is the voltage across the circuit (Volts)
• \(I\) is the current flowing through the circuit (Amperes)
• \(R\) is the resistance (Ohms)

In any practical setting, knowing Ohm's Law allows you to solve for any of the three variables if the other two are known.
How does this relate to temperature and resistance? Changes in temperature affect resistance. By using the temperature coefficient, you can determine the new resistance at a given temperature and then apply Ohm’s Law to find out how voltage and current will be affected.

Thermal effect on resistivity

Resistivity is a fundamental property of materials that affects their electrical resistance. When we talk about the thermal effect on resistivity, we're referring to how heating a conductor, such as copper, impacts its ability to conduct electricity.
Here's what happens:

• **Thermal Agitation:** As the temperature increases, the atoms in the copper vibrate more intensely, which can impede the flow of electrons.
• **Increased Collisions:** The electrons face more interruptions in their flow, leading to increased resistance within the wire.

Consequently, copper's resistivity rises with temperature, influencing the overall resistance of the wire. Understanding these interactions helps engineers design more reliable systems that maintain efficiency despite temperature variations.