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Chapter 26: Problem 3

A resistor with \(R_{1}=25.0 \Omega\) is connected to a battery that has negligible internal resistance and electrical energy is dissipated by \(R_{1}\) at a rate of \(36.0 \mathrm{~W}\). If a second resistor with \(R_{2}=15.0 \Omega\) is connected in series with \(R_{1},\) what is the total rate at which electrical energy is dissipated by the two resistors?

Short Answer

Expert verified
The total rate at which electrical energy is dissipated by the two resistors is 57.6W.

Step by step solution

01

Find the current

First, we need to find the current. Given that the power dissipated by the first resistor \(P_{1}\) is 36.0W and its resistance \(R_{1}\) is 25.0惟, the current \(I\) can be found by rearranging the formula \(P_{1} = I^{2} * R_{1}\) to \(I = \sqrt { \frac {P_{1}} {R_{1}} }\), which gives us \(I = \sqrt { \frac {36.0W} {25.0 \Omega} }\) = 1.2A.
02

Find the total resistance

Next, add the values of \(R_{1}\) and \(R_{2}\) together to get the total resistance \(R_{T} = R_{1} + R_{2} = 25.0 \Omega + 15.0 \Omega = 40.0 \Omega.\)
03

Find the total power dissipated

Finally, using the current from Step 1 and the total resistance from Step 2, apply the power formula again \(P_{T} = I^{2} * R_{T}\) to get \(P_{T} = (1.2A)^{2} * 40.0 \Omega = 57.6W.\)

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Key Concepts

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

series circuit
When resistors are connected in a series circuit, the electrical current that flows through them is the same across each component.
This is because there is only one path for the current to take.
But the voltage across each resistor may be different.
  • In a series circuit, the total resistance is the sum of the resistances of each resistor.
  • This means you'll have more resistance compared to a single resistor.
  • Series circuits are commonly used when you want the same current to flow through all components.
For example, in the exercise where two resistors are connected in series, both resistors have the same current of 1.2 A flowing through them. The series connection ensures that the total resistance is simply the sum of the individual resistances: 25 惟 for the first resistor and 15 惟 for the second resistor, resulting in a total resistance of 40 惟.
resistor power dissipation
Resistor power dissipation is the process through which resistors convert electrical energy into heat energy. Power dissipation in a resistor is an important concept for understanding how resistors work and how much heat they generate.Power dissipation can be calculated using the formula:\[ P = I^2 * R \]where:
  • \( P \) is the power dissipated by the resistor
  • \( I \) is the current flowing through the resistor
  • \( R \) is the resistance of the resistor
The exercise gives us an example of power dissipation with a figure of 36W for a single resistor with 25 惟. With a current of 1.2 A, we later calculate that the two resistors together dissipate 57.6 W when in series.
When designing circuits, it's crucial to ensure that resistors can handle the amount of power dissipated to avoid overheating.
total resistance calculation
Calculating the total resistance in series circuits is straightforward and involves adding up the resistances of each resistor.
  • This is because the total resistance in a series circuit is the sum total of all individual resistances.
  • It reflects the cumulative opposition to the flow of current through the circuit.
In the given problem, for instance, we have two resistors: one with 25 惟 and another with 15 惟. The total resistance is calculated as:\[ R_T = R_1 + R_2 = 25 \, \Omega + 15 \, \Omega = 40 \, \Omega \]Understanding this concept allows you to easily predict how the circuit behavior changes when you add more resistors in series, as it increases the total resistance, reducing the current if the voltage remains constant.

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Most popular questions from this chapter

The power rating of a resistor is the maximum power the resistor can safely dissipate without too great a rise in temperature and hence damage to the resistor. (a) If the power rating of a \(15 \mathrm{k} \Omega\) resistor is \(5.0 \mathrm{~W},\) what is the maximum allowable potential difference across the terminals of the resistor? (b) \(A\) \(9.0 \mathrm{k} \Omega\) resistor is to be connected across a \(120 \mathrm{~V}\) potential difference. What power rating is required? (c) \(\mathrm{A} 100.0 \Omega\) and a \(150.0 \Omega\) resistor, both rated at \(2.00 \mathrm{~W}\), are connected in series across a variable potential difference. What is the greatest this potential difference can be without overheating either resistor, and what is the rate of heat generated in each resistor under these conditions?

A \(1500 \mathrm{~W}\) electric heater is plugged into the outlet of a \(120 \mathrm{~V}\) circuit that has a 20 A circuit breaker. You plug an electric hair dryer into the same outlet. The hair dryer has power settings of \(600 \mathrm{~W}, 900 \mathrm{~W}\), \(1200 \mathrm{~W}\), and \(1500 \mathrm{~W}\). You start with the hair dryer on the \(600 \mathrm{~W}\) setting and increase the power setting until the circuit breaker trips. What power setting caused the breaker to trip?

Assume that a typical open ion channel spanning an axon鈥檚 membrane has a resistance of \(1 \times 10^{11} \Omega .\) We can model this ion channel, with its pore, as a 12-nm-long cylinder of radius \(0.3 \mathrm{nm}\). What is the resistivity of the fluid in the pore? (a) \(10 \Omega \cdot \mathrm{m}\) (b) \(6 \Omega \cdot \mathrm{m}\) (c) \(2 \Omega \cdot \mathrm{m} ;\) (d) \(1 \Omega \cdot \mathrm{m}\)

A \(1.50 \mu \mathrm{F}\) capacitor is charging through a \(12.0 \Omega\) resistor using a \(10.0 \mathrm{~V}\) battery. What will be the current when the capacitor has acquired \(\frac{1}{4}\) of its maximum charge? Will it be \(\frac{1}{4}\) of the maximum current?

An emf source with \(\mathcal{E}=120 \mathrm{~V},\) a resistor with \(R=80.0 \Omega\) and a capacitor with \(C=4.00 \mu \mathrm{F}\) are connected in series. As the capacitor charges, when the current in the resistor is \(0.900 \mathrm{~A},\) what is the magnitude of the charge on each plate of the capacitor?
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