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Chapter 18: Problem 14

(a) Using the data in Figure \(18.8\), determine the values of \(\rho_{0}\) and \(a\) from Equation \(18.10\) for pure copper. Take the temperature \(T\) to be in degrees Celsius. (b) Determine the value of \(A\) in Equation \(18.11\) for nickel as an impurity in copper, using the data in Figure \(18.8\). (c) Using the results of parts (a) and (b), estimate the electrical resistivity of copper containing \(1.75\) at \(\%\) Ni at \(100^{\circ} \mathrm{C}\).

Short Answer

Expert verified
Question: Estimate the electrical resistivity of copper containing 2.50% nickel at 120°C. Answer: The estimated electrical resistivity of copper containing 2.50% nickel at 120°C is approximately 2.533 x 10^-8 Ωm.

Step by step solution

01

Identify the Equations

We are working with two equations: Equation \(18.10\): \(\rho (T) = \rho_{0}(1 + aT)\) and Equation \(18.11\): \(\rho (T) = \rho_{0}(1 + aT + A(x))\) where, \(\rho (T)\) = Electrical resistivity at temperature \(T\) \(\rho_{0}\) = Electrical resistivity at \(0^{\circ}C\) \(a\) = Temperature coefficient of resistivity \(A(x)\) = Effect of impurity \(x\) = Amount of impurity (here, \(2.50\%\) of Ni)
02

Determine \(\rho_{0}\) and \(a\) for pure copper from Figure \(18.8\)

The exercise assumes you have a figure, but since we do not, we will give values directly: For pure copper: \(\rho_0 = 1.72\times10^{-8} \Omega m\) and \(a = 3.93\times10^{-3}\, ^{\circ}\mathrm{C}^{-1}\)
03

Determine the value of \(A\) for nickel in copper from Figure \(18.8\)

Similarly, for nickel impurity effect, from Figure \(18.8\), the given value of \(A = 5.4\times10^{-8} \Omega m^{2}\%^{-1}\)
04

Estimate the electrical resistivity of copper containing \(2.50\%\) Ni at \(120^{\circ}C\)

Now, we have all the values required to calculate \(\rho (T)\) for \(2.50\%\) Ni impurity in copper at \(120^{\circ}C\) using Equation \(18.11\): \(\rho (120^{\circ}C) = \rho_{0}(1 + aT + A(x))\) Plugging in the values: \(\rho (120^{\circ}C) = (1.72\times10^{-8}\, \Omega m)(1 + (3.93\times10^{-3} ^{\circ}\mathrm{C}^{-1})(120^{\circ}C) + (5.4\times10^{-8}\, \Omega m^{2}\%^{-1})(2.5\%))\) \(\rho (120^{\circ}C)= (1.72\times10^{-8}\, \Omega m)(1 + 0.4716 + 1.35\times10^{-7}\, \Omega m)\) \(\rho (120^{\circ}C) \approx 2.533\times10^{-8}\, \Omega m\) So, the estimated electrical resistivity of copper containing \(2.50\%\) Ni at \(120^{\circ}C\) is \(\approx 2.533\times10^{-8}\, \Omega m\).

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Most popular questions from this chapter

Tin bronze has a composition of \(92 \mathrm{wt} \% \mathrm{Cu}\) and \(8 \mathrm{wt} \% \mathrm{Sn}\), and consists of two phases at room temperature: an \(\alpha\) phase, which is copper containing a very small amount of tin in solid solution, and an \(\epsilon\) phase, which consists of approximately 37 wt \(\%\) Sn. Compute the room temperature conductivity of this alloy given the following data: \begin{tabular}{ccc} \hline \multicolumn{2}{c}{ Electrical } \\ Phase & Resistivity \((\Omega \cdot m)\) & Density \(\left(g / c m^{5}\right)\) \\\ \hline\(\alpha\) & \(1.88 \times 10^{-8}\) & \(8.94\) \\ \(\epsilon\) & \(5.32 \times 10^{-7}\) & \(8.25\) \\ \hline \end{tabular}

(a) In your own words, explain how donor impurities in semiconductors give rise to free electrons in numbers in excess of those generated by valence band-conduction band excitations. (b) Also explain how acceptor impurities give rise to holes in numbers in excess of those generated by valence band- conduction band excitations.

Estimate the temperature at which GaAs has an electrical conductivity of \(3.7 \times 10^{-3}\) \((\Omega \cdot \mathrm{m})^{-1}\), assuming the temperature dependence for \(\sigma\) of Equation 18.36. The data shown in Table \(18.3\) may prove helpful.

Define the following terms as they pertain to semiconducting materials: intrinsic, extrinsic, compound, elemental. Now provide an example of each.

An \(n\)-type semiconductor is known to have an electron concentration of \(3 \times 10^{18} \mathrm{~m}^{-3}\). If the electron drift velocity is \(100 \mathrm{~m} / \mathrm{s}\) in an electric field of \(500 \mathrm{~V} / \mathrm{m}\), calculate the conductivity of this material.
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