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Chapter 12: Problem 1777

The resistance of the wire made of silver at \(27^{\circ} \mathrm{C}\) temperature is equal to \(2.1 \Omega\) while at \(100^{\circ} \mathrm{C}\) it is \(2.7 \Omega\) calculate the temperature coefficient of the resistivity of silver. Take the reference temperature equal to \(20^{\circ} \mathrm{C}\) (A) \(4.02 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\) (B) \(0.402 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\) (C) \(40.2 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\) (D) \(4.02 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\)

Short Answer

Expert verified
The temperature coefficient of resistivity of silver is approximately \(4.02 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\).

Step by step solution

01

List the given information

We are provided with the following values: - Resistance at \(27^{\circ}\mathrm{C}\): \(R_{27} = 2.1 \Omega\) - Resistance at \(100^{\circ}\mathrm{C}\): \(R_{100} = 2.7 \Omega\) - Reference temperature: \(t_0 = 20^{\circ}\mathrm{C}\)
02

Set up two equations using the resistance formula

We can set up two equations using the temperature coefficient of resistivity formula: - Equation 1: \(R_{27} = R_0[1 + \alpha (27 - 20)]\) - Equation 2: \(R_{100} = R_0[1 + \alpha (100 - 20)]\)
03

Solve for R_0 using Equation 1

Using Equation 1, we can solve for the reference resistance \(R_0\): \(2.1 = R_0[1 + \alpha (7)]\)
04

Substitute R_0 from Equation 1 into Equation 2

Now we substitute the expression for \(R_0\) from Step 3 into Equation 2: \(2.7 = \frac{2.1}{1 + 7\alpha}[1 + 80\alpha]\)
05

Solve for alpha

Now, we need to solve the equation for \(\alpha\): \(\frac{2.7}{1 + 80\alpha} = \frac{2.1}{1 + 7\alpha}\) Cross-multiplying, we get: \(2.7 (1 + 7\alpha) = 2.1 (1 + 80\alpha)\) Expanding both sides: \(2.7 + 18.9\alpha = 2.1 + 168\alpha\) Now we can isolate \(\alpha\): \(150.9\alpha = 0.6\) \(\alpha = \frac{0.6}{150.9}\)
06

Calculate the value of alpha and choose the correct option

Now, we can calculate the value of \(\alpha\): \(\alpha \approx 0.00402^{\circ}\mathrm{C}^{-1}\) Now, we can see that the correct option is: (A) \(4.02 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\)

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Most popular questions from this chapter

In an experiment to measure the internal resistance of a cell by a potentiometer it is found that all the balance points at a length of $2 \mathrm{~m}$ when the cell is shunted by a 5 ohm resistance and is at a length of \(3 \mathrm{~m}\) when the cell is shunted by a 10 ohm resistance, the internal resistance of the cell is then: (A) \(1.5 \Omega\) (B) \(10 \Omega\) (C) \(15 \Omega\) (D) \(1 \Omega\)

In a wheat stone's bridge, three reststance \(P, Q\) and \(R\) connected in three arm a and the fourth arm is formed by two resistances \(\mathrm{S}_{1}\) and \(\mathrm{S}_{2}\) connected in parallel The condition for bridge to be balanced will be. (A) $(\mathrm{P} / \mathrm{Q})=\left\\{\mathrm{R} /\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\right\\}$ (B) $(\mathrm{P} / \mathrm{Q})=\left\\{(2 \mathrm{R}) /\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\right\\}$ (C) $(\mathrm{P} / \mathrm{Q})=\left[\left\\{\mathrm{R}\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\right\\} /\left\\{\mathrm{S}_{1} \mathrm{~S}_{2}\right\\}\right]$ (D) $(\mathrm{P} / \mathrm{Q})=\left[\left\\{\mathrm{R}\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\right\\} /\left\\{2 \mathrm{~S}_{1} \mathrm{~S}_{2}\right\\}\right]$

Resistance of a wire at \(50^{\circ} \mathrm{C}\) is \(5 \Omega\), and at \(100^{\circ} \mathrm{C}\) it is \(6 \Omega\) find its resistance at $0^{\circ} \mathrm{C}$ (A) \(4 \Omega\) (B) \(3 \Omega\) (C) \(2 \Omega\) (D) \(1 \Omega\)

The masses of three wires of copper are in the ratio of \(1: 3: 5\) and their lengths are in the ratio of \(5: 3: 1\). The ratio of their electrical resistance is: (A) \(1: 1: 1\) (B) \(1: 3: 5\) (C) \(5: 3: 1\) (D) \(125: 15: 1\)

An electric kettle has two coils when one of these is switched on. the water in the kettle boils in 6 minutes. When the other coil is switched on, boils in 3 minutes If the two coils are connected in parallel, the time taken to boil water in the kettle is. (A) 3 minutes (B) 6 minutes (C) 2 minutes (D) 9 minutes
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