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Chapter 22: Problem 44

A vacuum cleaner is plugged into a \(120.0-\mathrm{V}\) socket and uses 3.0 \(\mathrm{A}\) of current in normal operation when the back emf generated by the electric motor is 72.0 \(\mathrm{V}\) . Find the coil resistance of the motor.

Short Answer

Expert verified
The coil resistance is 16.0 惟.

Step by step solution

01

Understanding Circuit Basics

To solve the problem, we must understand that the total voltage ( 120.0 V) powers the vacuum cleaner, while the back electromotive force (back emf, 72.0 V) opposes the current flow. The effective voltage ( V_{ _eff}) driving the current through the coil is the difference between the applied voltage and back emf.
02

Calculate Effective Voltage

Subtract the back emf from the total voltage to determine the effective voltage. This is calculated as:\[ V_{\text{eff}} = 120.0\, \text{V} - 72.0\, \text{V} = 48.0\, \text{V}. \]
03

Use Ohm's Law for Resistance

Ohm's Law relates voltage, current, and resistance as \(V = IR\). We can rearrange this to solve for resistance: \( R = \frac{V}{I} \). Using the effective voltage and the provided current, we solve for coil resistance:
04

Calculate Coil Resistance

Using the formula from Ohm's Law, substitute \( V_{\text{eff}} = 48.0\, \text{V} \) and \( I = 3.0\, \text{A} \) to find the resistance:\[ R = \frac{48.0\, \text{V}}{3.0\, \text{A}} = 16.0\, \Omega. \]

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Key Concepts

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

Effective Voltage
When dealing with electrical circuits, the concept of effective voltage is crucial for understanding how devices operate. Think of it as the actual voltage that powers a device after accounting for all opposing forces, like the back electromotive force (back emf) in motors. To find the effective voltage in a circuit:
  • Start by identifying the total voltage supplied to the circuit, like the 120.0 V in our example.
  • Next, determine the opposing voltage, which in this context is the back emf 鈥� here, it鈥檚 72.0 V.
  • Subtract the back emf from the total voltage to obtain the effective voltage. In this case, it is 48.0 V as calculated by \( V_{\text{eff}} = 120.0\, \text{V} - 72.0\, \text{V} = 48.0\, \text{V} \).
This process helps you evaluate the true driving force of the current in the device's coil, which is essential for further calculations like finding the resistance.
Coil Resistance
Understanding coil resistance is central to diagnosing electrical devices, especially motors. Coil resistance is a measure of how much a coil resists electrical current. Here鈥檚 how it factors into device operation:
  • The resistance can be determined through Ohm鈥檚 Law, represented as \( V = IR \), where \( V \) is the effective voltage, \( I \) is the current, and \( R \) is the resistance.
  • Rearranging Ohm鈥檚 Law gives us the formula needed to find resistance: \( R = \frac{V}{I} \).
  • In the given exercise, using the calculated effective voltage of 48.0 V and the current of 3.0 A, we find the coil resistance to be 16.0 惟. In mathematical terms, \( R = \frac{48.0\, \text{V}}{3.0\, \text{A}} = 16.0\, \Omega \).
Coil resistance helps understand the efficiency and potential power loss in a motor, enabling you to identify if a device is operating optimally or needs maintenance.
Back Electromotive Force (emf)
Back electromotive force, often abbreviated as back emf, plays a significant role in electric motors. It鈥檚 the counter-voltage generated by the motor as it spins, opposing the source voltage.
  • Back emf is an inherent characteristic of motors since they act somewhat like generators when spinning.
  • In our vacuum cleaner exercise, the back emf is 72.0 V. This is the voltage generated while the motor's armature (the spinning part) cuts through the magnetic field lines.
  • This opposing force is crucial for controlling the motor's speed and efficiency, as it regulates the overall current flowing through the motor.
By understanding back emf, you gain insight into the motor's actual workload, allowing for better prediction, maintenance, and efficiency improvements in motor-driven devices.

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Most popular questions from this chapter

Multiple-Concept Example 13 reviews the concepts used in this problem. A long solenoid (cross-sectional area \(=1.0 \times 10^{-6} \mathrm{m}^{2}\) , number of turns per unit length \(=2400\) turns/m) is bent into a circular shape so it looks like a donut. This wire-wound donut is called a toroid. Assume that the diameter of the solenoid is small compared to the radius of the toroid, which is 0.050 \(\mathrm{m}\) . Find the emf induced in the toroid when the current decreases to 1.1 A from 2.5 A in a time of 0.15 s.

ssm A piece of copper wire is formed into a single circular loop of radius 12 \(\mathrm{cm} .\) A magnetic field is oriented parallel to the normal to the loop, and it increases from 0 to 0.60 \(\mathrm{T}\) in a time of 0.45 s. The wire has a resistance per unit length of \(3.3 \times 10^{-2} \Omega / \mathrm{m} .\) What is the average electrical energy dissipated in the resistance of the wire?

ssm The plane of a flat, circular loop of wire is horizontal. An external magnetic field is directed perpendicular to the plane of the loop. The magnitude of the external magnetic field is increasing with time. Because of this increasing magnetic field, an induced current is flowing clockwise in the loop, as viewed from above. What is the direction of the external magnetic field? Justify your conclusion.

ssm Multiple-Concept Example 13 reviews some of the principles used in this problem. Suppose you wish to make a solenoid whose self-inductance is 1.4 \(\mathrm{mH}\) . The inductor is to have a cross-sectional area of \(1.2 \times 10^{-3} \mathrm{m}^{2}\) and a length of 0.052 \(\mathrm{m}\) . How many turns of wire are needed?

mmh The maximum strength of the earth's magnetic field is about \(6.9 \times 10^{-5} \mathrm{T}\) near the south magnetic pole. In principle, this field could be used with a rotating coil to generate \(60.0 . \mathrm{-Hz}\) ac electricity. What is the minimum number of turns (area per turn \(=0.022 \mathrm{m}^{2}\) ) that the coil must have to produce an rms voltage of 120 \(\mathrm{V}\) ?
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