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Chapter 21: Problem 71

The field coils used in an ac motor are designed to have a resistance of $0.45 \Omega\( and an impedance of \)35.0 \Omega$ What inductance is required if the frequency of the ac source is (a) \(60.0 \mathrm{Hz} ?\) and (b) $0.20 \mathrm{kHz} ?$

Short Answer

Expert verified
Answer: (a) 1.66 H (b) 0.055 H

Step by step solution

01

Calculate the inductive reactance at each frequency

First, we need to calculate the inductive reactance (\(X_L\)) for both frequencies using the formula: $$X_L = \sqrt{Z^2 - R^2}$$ For (a): $$X_{L1} = \sqrt{(35.0\,\Omega)^2 - (0.45\,\Omega)^2}$$ For (b): $$X_{L2} = \sqrt{(35.0\,\Omega)^2 - (0.45\,\Omega)^2}$$ Notice that the given impedance and resistance are the same for both frequencies, so our calculation for both \(X_{L1}\) and \(X_{L2}\) is the same.
02

Calculate the inductance for each frequency

Now that we have the inductive reactance at each frequency, we can use the formula that relates frequency, inductance, and inductive reactance: $$X_L = 2\pi f L$$ We will solve for \(L\) at each frequency: For (a): $$L_1 = \frac{X_{L1}}{2\pi f_1}$$ For (b): $$L_2 = \frac{X_{L2}}{2\pi f_2}$$
03

Plug in the values and compute the results

Now we will plug in the values for the inductive reactance and frequency, and compute the inductance for both cases: For (a): $$L_1 = \frac{\sqrt{(35.0\,\Omega)^2 - (0.45\,\Omega)^2}}{2\pi (60.0\,\mathrm{Hz})}$$ For (b): $$L_2 = \frac{\sqrt{(35.0\,\Omega)^2 - (0.45\,\Omega)^2}}{2\pi (0.20\,\mathrm{kHz})}$$ Solving for \(L_1\) and \(L_2\), we get the required inductance for each frequency: $$L_1 = 1.66 \,\mathrm{H}$$ $$L_2 = 0.055 \,\mathrm{H}$$

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Most popular questions from this chapter

What is the reactance of an air core solenoid of length \(8.0 \mathrm{cm},\) radius \(1.0 \mathrm{cm},\) and 240 turns at a frequency of $15.0 \mathrm{kHz} ?$
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A 22 -kV power line that is \(10.0 \mathrm{km}\) long supplies the electric energy to a small town at an average rate of \(6.0 \mathrm{MW} .\) (a) If a pair of aluminum cables of diameter \(9.2 \mathrm{cm}\) are used, what is the average power dissipated in the transmission line? (b) Why is aluminum used rather than a better conductor such as copper or silver?
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