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Chapter 12: Problem 1744

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

Expert verified
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

01

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.
02

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: \(R_3 = R_0 (1 + α(t - t_0))\)
03

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 \(α = 4 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\): \(3R_0 = R_0 (1 + (4 \times 10^{-3})(t - 0))\)
04

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) \(500^{\circ} \mathrm{C}\).

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