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Chapter 20: Problem 50

A coffee-maker \((14 \Omega)\) and a toaster \((19 \Omega)\) are connected in parallel to the same \(120-\mathrm{V}\) outlet in a kitchen. How much total power is supplied to the two appliances when both are turned on?

Short Answer

Expert verified
1786.46 W

Step by step solution

01

Understand the Setup

Both the coffee-maker and the toaster are connected in parallel. In parallel circuits, the voltage across each component is the same and equal to the source voltage. Here, it's 120 V.
02

Calculate Resistance and Power for Each Appliance

The resistance of the coffee-maker is given as 14 惟 and that of the toaster as 19 惟. The power consumed by a single appliance connected to a voltage source can be calculated using the formula: \[ P = \frac{V^2}{R} \]where \( P \) is the power, \( V \) is the voltage, and \( R \) is the resistance.
03

Calculate Power for the Coffee-Maker

Using the formula from Step 2, \( P_{coffee} = \frac{120^2}{14} \). Calculate the value:\[ P_{coffee} = \frac{14400}{14} = 1028.57 \, \text{W} \]
04

Calculate Power for the Toaster

Similarly, for the toaster, \( P_{toaster} = \frac{120^2}{19} \). Calculate the value:\[ P_{toaster} = \frac{14400}{19} = 757.89 \, \text{W} \]
05

Calculate Total Power Supplied

Since the appliances are in parallel, the total power is the sum of the powers of each appliance:\[ P_{total} = P_{coffee} + P_{toaster} = 1028.57 + 757.89 \]
06

Sum the Powers

Add the power values:\[ P_{total} = 1786.46 \, \text{W} \]

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Key Concepts

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

Resistance Calculation
When dealing with parallel circuits, resistance calculation becomes much more than just dealing with numbers. In a parallel circuit, each component, like our coffee-maker and toaster, has its resistances working separately.
  • The resistance across any component is the opposite of the straightforward sum; it requires inversions because the current has multiple paths.
  • In this particular exercise, we first look at the individual resistances as given: 14 惟 for the coffee-maker and 19 惟 for the toaster.
  • While combining resistances in parallel isn't directly applicable here since we're looking at powers, it's important to mention the rule: \( \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots \).
Ultimately, understanding resistances helps us see how they govern current flow and power usage in any circuit component.
Power Calculation
Power calculation in circuits is crucial for understanding how much energy each device consumes. When dealing with a uniform voltage supply (like 120 V in our kitchen example), power can be calculated with the formula: \[ P = \frac{V^2}{R} \]
For our coffee-maker:
  • We substitute 120 V and 14 惟 into our formula: \( P_{coffee} = \frac{120^2}{14} = 1028.57 \text{ W} \).
  • This calculation shows the appliance's energy requirement in watts, a direct measurement of its power consumption.
Similarly, our toaster:
  • Using 120 V and 19 惟: \( P_{toaster} = \frac{120^2}{19} = 757.89 \text{ W} \).
  • Add the power values to find total energy necessary when both are in operation: \( P_{total} = 1786.46 \text{ W} \).
This approach gives clear insights into how appliances draw power individually, as well as collectively when operating in parallel.
Ohm's Law
Ohm's Law is a foundational principle in electronics, stating that voltage (\( V \)) is the product of current (\( I \)) and resistance (\( R \)): \( V = IR \). This relationship helps us comprehend the interaction between voltage, current, and resistance across a circuit component.
  • In the context of parallel circuits, each component experiences the same voltage, allowing different currents through each path depending on their resistance.
  • When applied to our example, understanding this law helps to predict how the different resistances of the coffee-maker and toaster react to the consistent 120 V supply.
  • The inversely proportional relationship between resistance and current flow indicates that lower resistance allows more current, leading to higher power usage as observed with the coffee-maker.
Thus, appreciating Ohm's Law provides deeper insight into why the coffee-maker draws more power than the toaster at the same voltage level.

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Most popular questions from this chapter

A digital thermometer employs a thermistor as the temperaturesensing element. A thermistor is a kind of semiconductor and has a large negative temperature coefficient of resistivity \(\alpha\) . Suppose that \(\alpha=-0.060\left(\mathrm{C}^{\circ}\right)^{-1}\) for the thermistor in a digital thermometer used to measure the temperature of a patient. The resistance of the thermistor decreases to 85\(\%\) of its value at the normal body temperature of \(37.0^{\circ} \mathrm{C}\) . What is the patient's temperature?

Three resistors are connected in series across a battery. The value of each resistance and its maximum power rating are as follows: 2.0\(\Omega\) and \(4.0 \mathrm{W}, 12.0 \Omega\) and \(10.0 \mathrm{W},\) and 3.0\(\Omega\) and 5.0 \(\mathrm{W}\) . (a) What is the greatest voltage that the battery can have without one of the resistors burning up? (b) How much power does the battery deliver to the circuit in ( a \()\) ?

Two wires have the same cross-sectional area and are joined end to end to form a single wire. One is tungsten, which has a temperature coefficient of resistivity of \(\alpha=0.0045\left(\mathrm{C}^{\circ}\right)^{-1}\) . The other is carbon, for which \(\alpha=-0.0005\left(\mathrm{C}^{\circ}\right)^{-1} .\) The total resistance of the composite wire is the sum of the resistances of the pieces. The total resistance of the composite does not change with temperature. What is the ratio of the lengths of the tungsten and carbon sections? Ignore any changes in length due to thermal expansion.

A piece of Nichrome wire has a radius of \(6.5 \times 10^{-4} \mathrm{m} .\) It is used in a laboratory to make a heater that uses \(4.00 \times 10^{2} \mathrm{W}\) of power when connected to a voltage source of 120 \(\mathrm{V}\) . Ignoring the effect of temperature on resistance, estimate the necessary length of wire.

The current in a \(47-\Omega\) resistor is 0.12 A. This resistor is in series with a \(28-\Omega\) resistor, and the series combination is connected across a battery. What is the battery voltage?
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