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Chapter 10: Problem 19

At what temperature will the resistance of a copper wire become three times its value at \(0^{\circ} \mathrm{C}\) (Temperature coefficient of resistance for copper \(=4 \times 10^{-3} /{ }^{\circ} \mathrm{C}\) ) (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 becomes three times its value at \(0^{\circ} \mathrm{C}\) is (C) \(500^{\circ} \mathrm{C}\).

Step by step solution

01

Write the formula for change in resistance with temperature

The formula for the change in resistance with temperature is: \[R_t = R_0 ( 1 + \alpha ( t - 0))\] Where \(R_t\) is the resistance at temperature t, \(R_0\) is the resistance at \(0^{\circ} \mathrm{C}\), \(\alpha\) is the temperature coefficient of resistance, and t is the temperature in Celsius. We want to find t when \(R_t\) is three times the resistance at \(0^{\circ} \mathrm{C}\), or in other words, \(R_t = 3R_0\).
02

Plug in the given values and re-write the equation

We know that \(R_t = 3R_0\) and \(\alpha = 4 \times 10^{-3} /{ }^{\circ} \mathrm{C}\). Plugging these values into the formula, we get: \[3R_0 = R_0 (1 + 4 \times 10^{-3} (t - 0))\]
03

Solve for temperature (t)

We first simplify the equation and then solve for t: \[3 = 1 + 4 \times 10^{-3} t\] \[t = \frac{2}{4 \times 10^{-3}}\] \[t = 500^{\circ} \mathrm{C}\]
04

Choose the answer

Since the temperature required for the resistance of the copper wire to become three times its value at \(0^{\circ} \mathrm{C}\) is \(500^{\circ} \mathrm{C}\), the correct answer is: (C) \(500^{\circ} \mathrm{C}\)

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Most popular questions from this chapter

1000 drops of a liquid of surface tension \(\sigma\) and radius \(r\) join together to form a big single drop. The energy released raises the temperature of the drop. If \(\rho\) be the density of the liquid and \(S\) be the specific heat, the rise in temperature of the drop would be \((J=\) Joule's equivalent of heat) (A) \(\frac{\sigma}{J r S \rho}\) (B) \(\frac{10 \sigma}{J r S \rho}\) (C) \(\frac{100 \sigma}{J r S \rho}\) (D) \(\frac{27 \sigma}{10 J_{r} S \rho}\)

A solid body of constant heat capacity \(1 \mathrm{~J} /{ }^{\circ} \mathrm{C}\) is being heated by keeping it in contact with reservoirs in two ways: (i) Sequentially keeping in contact with 2 reservoirs such that each reservoir supplies same amount of heat. (ii) Sequentially keeping in contact with 8 reservoirs such that each reservoir supplies same amount of heat. In both the case body is brought from initial temperature \(100^{\circ} \mathrm{C}\) to final temperature \(200^{\circ} \mathrm{C}\). Entropy change of the body in the two cases respectively is \([2015]\) (A) \(\ln 2, \ln 2\) (B) \(\ln 2,2 \ln 2\) (C) \(2 \ln 2,8 \ln 2\) (D) \(\ln 2,4 \ln 2\)

The ratio of coefficients of cubical expansion and linear expansion is (A) \(1: 1\) (B) \(3: 1\) (C) \(2: 1\) (D) None of these

The temperature of a substance increases by \(27^{\circ} \mathrm{C}\). On the Kelvin scale, this increase is equal to (A) \(300 \mathrm{~K}\) (B) \(2.46 \mathrm{~K}\) (C) \(27 \mathrm{~K}\) (D) \(7 \mathrm{~K}\)

The molar specific heats of an ideal gas at constant pressure and volume are denoted by \(C_{p}\) and \(C_{p}\), respectively. Further, \(\frac{C_{p}}{C_{v}}=\gamma\) and \(R\) is the gas constant for 1 gm mole of a gas. Then \(C_{v}\) is equal to (A) \(R\) (B) \(\gamma R\) (C) \(\frac{R}{\gamma-1}\) (D) \(\frac{\gamma R}{\gamma-1}\)
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