/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 Air Conditioner An air condition... [FREE SOLUTION] | 91影视

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Chapter 33: Problem 67

Air Conditioner An air conditioner connected to a \(120 \mathrm{~V} \mathrm{rms}\) ac line is equivalent to a \(12.0 \Omega\) resistance and a \(1.30 \Omega\) inductive reactance in series. (a) Calculate the impedance of the air conditioner. (b) Find the average rate at which energy is supplied to the appliance.

Short Answer

Expert verified
The impedance is approximately 12.1 惟. The average power supplied is 1176.12 W.

Step by step solution

01

- Identify Given Values

Given: - RMS Voltage, \( V_{rms} = 120 \text{ V} \) - Resistance, \( R = 12.0 \Omega \) - Inductive Reactance, \( X_L = 1.30 \Omega \)
02

- Calculate Impedance

Impedance in an RL circuit is given by: \[ Z = \sqrt{R^2 + X_L^2} \] Substitute the given values: \[ Z = \sqrt{(12.0)^2 + (1.30)^2} = \sqrt{144 + 1.69} = \sqrt{145.69} \approx 12.1 \Omega \]
03

- Find the Current

The current through the appliance can be found using Ohm鈥檚 Law: \[ I = \frac{V_{rms}}{Z} \] \[ I = \frac{120 \text{ V}}{12.1 \Omega} \approx 9.9 \text{ A} \]
04

- Calculate Average Power Supplied

The average power supplied to the appliance can be calculated using: \[ P_{avg} = I^2 R \] Substitute the current and resistance values: \[ P_{avg} = (9.9 \text{ A})^2 \times 12.0 \Omega = 98.01 \text{ A}^2 \times 12.0 \Omega = 1176.12 \text{ W} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

impedance in RL circuits
When dealing with circuits containing both resistors (R) and inductors (L), we often encounter the concept of **impedance**. Impedance is a measure of how much a circuit resists the flow of alternating current (AC). It combines the effects of both resistance and inductive reactance.
The formula for calculating impedance (Z) in an RL circuit is: \[ Z = \sqrt{R^2 + X_L^2} \]
Here:
  • **R** is the resistance in ohms (惟).
  • **X_L** is the inductive reactance in ohms (惟).
In the given problem:
  • **R** is 12.0 惟
  • **X_L** is 1.30 惟
Plugging these values into the formula:
\[ Z = \sqrt{(12.0)^2 + (1.30)^2} = \sqrt{144 + 1.69} = \sqrt{145.69} \approx 12.1 惟 \]
This means the impedance of the air conditioner is approximately 12.1 惟.
Ohm's Law
Ohm's Law is a fundamental principle used in electrical engineering and physics. It states that the current (I) passing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R). The mathematical form of Ohm's Law is: \[ I = \frac{V}{R} \]
However, for AC circuits with impedance (Z), we modify the formula to: \[ I = \frac{V_{rms}}{Z} \]
Using the provided values:
  • RMS Voltage, **V_rms** is 120 V
  • Impedance, **Z** is 12.1 惟
Thus:
\[ I = \frac{120 \text{V}}{12.1 \Omega} \approx 9.9 \text{A} \]
This means the current flowing through the air conditioner is approximately 9.9 A.
average power calculation
The average power supplied to a device in an AC circuit can be calculated using the current and resistance. The formula used is: \[ P_{avg} = I^2 R \]
Where:
  • **I** is the current in amperes (A)
  • **R** is the resistance in ohms (惟)
From the previous section, we know:
  • Current, **I** is 9.9 A
  • Resistance, **R** is 12.0 惟
Now, substituting these values into the formula:
\[ P_{avg} = (9.9 \text{A})^2 \times 12.0 \Omega = 98.01 \text{A}^2 \times 12.0 \Omega = 1176.12 \text{W} \]
Therefore, the average power supplied to the air conditioner is approximately 1176.12 W.

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Most popular questions from this chapter

SHM A \(0.50 \mathrm{~kg}\) body oscillates in simple harmonic motion on a spring that, when extended \(2.0 \mathrm{~mm}\) from its equilibrium, has an \(8.0 \mathrm{~N}\) restoring force. (a) What is the angular frequency of oscillation? (b) What is the period of oscillation? (c) What is the capacitance of an \(L C\) circuit with the same period if \(L\) is chosen to be \(5.0 \mathrm{H}\) ?

Capacitive Reactance An ac generator with \(\mathscr{E}^{\max }=220 \mathrm{~V}\) and operating at \(400 \mathrm{~Hz}\) causes oscillations in a series \(R L C\) circuit having \(R=220 \Omega, L=150 \mathrm{mH}\), and \(C=24.0 \mu \mathrm{F}\). Find (a) the capacitive reactance \(X_{C},(\mathrm{~b})\) the impedance \(Z\), and \((\mathrm{c})\) the current amplitude \(I .\) A second capacitor of the same capacitance is then connected in series with the other components. Determine whether the values of (d) \(X_{C}\), (e) \(Z\), and (f) \(I\) increase, decrease, or remain the same.

Interstellar Space The magnetic field in the interstellar space of our galaxy has a magnitude of about \(10^{-10} \mathrm{~T}\). How much energy is stored in this field in a cube 10 light-years on edge? (For scale, note that the nearest star is \(4.3\) light-years distant and the radius of our galaxy is about \(8 \times 10^{4}\) light-years.)

Toroidal Inductor A toroidal inductor with an inductance of \(90.0 \mathrm{mH}\) encloses a volume of \(0.0200 \mathrm{~m}^{3}\). If the average energy density in the toroid is \(70.0 \mathrm{~J} / \mathrm{m}^{3}\), what is the current through the inductor?

What Is the Capacitance What is the capacitance of an oscillating \(L C\) circuit if the maximum charge on the capacitor is \(1.60 \mu \mathrm{C}\) and the total energy is \(140 \mu \mathrm{J}\) ?
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