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Chapter 22: Problem 38

A vacuum cleaner is plugged into a \(120.0-\mathrm{V}\) socket and uses 3.0 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 of the motor is \( 16.0 \Omega \).

Step by step solution

01

Understanding the Components

To find the coil resistance of the motor, we have to first understand the concept of back emf (electromotive force). Back emf is the voltage generated by the motor that opposes the supply voltage. It effectively reduces the total voltage across the motor's coil.
02

Applying Ohm's Law

Ohm's Law states that \( V = IR \), where \( V \) is voltage, \( I \) is current, and \( R \) is resistance. In this case, the total voltage (supply voltage) is \( 120.0 \text{ V} \), and the current is \( 3.0 \text{ A} \). However, we need the potential difference across the coil.
03

Calculate Potential Difference Across the Coil

The potential difference across the coil is equal to the supply voltage minus the back emf: \( V_{coil} = 120.0 \text{ V} - 72.0 \text{ V} = 48.0 \text{ V} \).
04

Finding Coil Resistance

Using Ohm's Law \( V_{coil} = I imes R \), we can find the resistance of the coil: \[ R = \frac{V_{coil}}{I} = \frac{48.0 \text{ V}}{3.0 \text{ A}} = 16.0 \Omega \].

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Key Concepts

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

Back EMF
When we talk about back electromotive force or back emf, we refer to a fundamental concept in the operation of electric motors. Back emf is a voltage that is induced in the motor's coil when it is spinning. This induced voltage works against the initial voltage supplied to the motor.
This opposing voltage is crucial for the control of the motor's speed and energy efficiency.
  • It effectively reduces the actual voltage across the motor's internal coil.
  • Back emf increases proportionally to the motor's speed.
  • When a motor starts from a stop, the back emf is initially zero because the motor is not yet spinning.
Understanding back emf is essential as it allows engineers to design motors that automatically adjust to the load or resistance they're encountering.
Coil Resistance
The coil resistance in an electric motor is equated to the resistance of the wire windings inside the motor. This resistance is an intrinsic property of the material, commonly copper, used for the coil.
  • Coil resistance affects how much current can flow through the motor.
  • Higher resistance means less current and vice versa, given a constant voltage.
  • It's an important factor in determining the efficiency and thermal characteristics of the motor.
In our exercise, by using Ohm's Law, which is given by the formula: \( V = IR \), we can isolate the coil's resistance using the known voltage across the coil (after accounting for back emf) and the current flowing through it.
Electric Motor
An electric motor is a device that uses electrical energy to produce mechanical motion. These machines are utilized in countless applications, from small devices like vacuum cleaners to large industrial machines.
  • Electric motors convert electrical energy into mechanical energy by creating rotational force or torque.
  • Motors rely on electromagnetic interactions between the coil and magnets to function.
  • The core components of an electric motor are the rotor, stator, windings, and commutator.
In the context of our exercise, understanding the interplay between supplied voltage, back emf, and coil resistance is crucial to assessing how effectively the motor can convert electrical energy into motion. This knowledge ensures that motors operate efficiently and safely across various operating conditions.

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Most popular questions from this chapter

Interactive Solution \(\underline{22.55}\) at offers one approach to problems such as this one. The secondary coil of a step-up transformer provides the voltage that operates an electrostatic air filter. The turns ratio of the transformer is 50: 1 . The primary coil is plugged into a standard \(120-\mathrm{V}\) outlet. The current in the secondary coil is \(1.7 \times 10^{-3} \mathrm{~A} .\) Find the power consumed by the air filter.

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?

A \(120.0\) - \(\mathrm{V}\) motor draws a current of \(7.00 \mathrm{~A}\) when running at normal speed. The resistance of the armature wire is \(0.720 \Omega\). (a) Determine the back emf generated by the motor. (b) What is the current at the instant when the motor is just turned on and has not begun to rotate? (c) What series resistance must be added to limit the starting current to \(15.0 \mathrm{~A} ?\)

The coil within an ac generator has an area per turn of \(1.2 \times 10^{-2} \mathrm{~m}^{2}\) and consists of 500 turns. The coil is situated in a 0.13-T magnetic field and is rotating at an angular speed of \(34 \mathrm{rad} / \mathrm{s}\). What is the emf induced in the coil at the instant when the normal to the loop makes an angle of \(27^{\circ}\) with respect to the direction of the magnetic field?

Concept Questions A constant current \(I\) exists in a solenoid whose inductance is \(L .\) The current is then reduced to zero in a certain amount of time. (a) If the wire from which the solenoid is made has no resistance, is there a voltage across the solenoid during the time when the current is constant? (b) If the wire from which the solenoid is made has no resistance, is there an emf across the solenoid during the time that the current is being reduced to zero? (c) Does the solenoid store electrical energy when the current is constant? If so, express this energy in terms of the current and the inductance. (d) When the current is reduced from its constant value to zero, what is the rate at which energy is removed from the solenoid? Express your answer in terms of the initial current, the inductance, and the time during which the current goes to zero. Problem A solenoid has an inductance of \(L=3.1 \mathrm{H}\) and carries a current of \(I=15 \mathrm{~A}\). (a) If the current goes from 15 to \(0 \mathrm{~A}\) in a time of \(75 \mathrm{~ms}\), what is the emf induced in the solenoid? (b) How much electrical energy is stored in the solenoid? (c) At what rate must the electrical energy be removed from the solenoid when the current is reduced to zero in \(75 \mathrm{~ms} ?\)
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