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Chapter 19: Problem 36

A \(500 \Omega\) and a \(200 \Omega\) resistor are connected in series with an ideal battery that has an emf of \(20 \mathrm{~V}\). (a) What current flows through each resistor? (b) What power is delivered to each resistor? (c) What power is supplied by the battery?

Short Answer

Expert verified
(a) 0.02857 A; (b) 0.4081 W and 0.1632 W; (c) 0.5714 W.

Step by step solution

01

Calculate Total Resistance

When resistors are connected in series, the total resistance is the sum of the individual resistances. Here, the total resistance \( R_{total} \) is \( 500 \Omega + 200 \Omega = 700 \Omega \).
02

Determine Total Current Using Ohm's Law

Apply Ohm's Law, \( V = IR \), to find the current \( I \). The total voltage \( V = 20 \) V and \( R_{total} = 700 \Omega \). Thus, \( I = \frac{V}{R_{total}} = \frac{20}{700} = 0.02857 \) A.
03

Verify Current Through Each Resistor

In a series circuit, the same current flows through each resistor, so the current through each resistor is \( 0.02857 \) A.
04

Calculate Power for Each Resistor

Use the formula \( P = I^2R \) to find the power for each resistor. \( P_{500} = (0.02857)^2 \times 500 = 0.4081 \) W for the \(500 \Omega\) resistor, and \( P_{200} = (0.02857)^2 \times 200 = 0.1632 \) W for the \(200 \Omega\) resistor.
05

Calculate Total Power Supplied by Battery

The power supplied by the battery is given by \( P = IV \), where \( I = 0.02857 \) A and \( V = 20 \) V. \( P = 0.02857 \times 20 = 0.5714 \) W.

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Key Concepts

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

Series Circuit
A series circuit is an electrical circuit where components are connected end-to-end in a single path. In this type of circuit, the same electric current flows through each device or component.
This is due to the fact that there is only one path for the current to travel. If one component in the circuit fails or is disconnected, the current flow stops throughout the entire circuit.
Key characteristics of a series circuit include:
  • All components share the same current.
  • Total resistance is the sum of individual resistances (R = R鈧� + R鈧� + ...).
  • Voltage across the circuit is divided among the components.
Understanding series circuits is foundational when analyzing how different resistors affect the flow of electricity.
Electric Power
Electric power refers to the rate at which electrical energy is transferred by an electric circuit. It is usually measured in watts (W). Electric power can be calculated in a circuit using various formulas, but a common one is:
\[ P = IV \]
where \( P \) is the power in watts, \( I \) is the current in amperes, and \( V \) is the voltage in volts.
Alternatively, power can also be determined using the formula:
  • \( P = I^2 R \)
  • \( P = \frac{V^2}{R} \)
These formulas indicate how power in a circuit changes depending on current and resistance. In the example exercise, power is calculated for each resistor to understand how energy is distributed.
Current Calculation
Current calculation is an essential part of analyzing electrical circuits. Ohm's Law provides a simple formula for this purpose:
\[ V = IR \]
Where \( V \) is voltage, \( I \) is current, and \( R \) is resistance. By rearranging this formula, you can solve for any one of these variables if the other two are known.
To find the current in a series circuit, you divide the total voltage by the total resistance:
  • \( I = \frac{V}{R_{total}} \)
This allows you to determine the current that flows through each component, as seen in the given solution. Understanding how to calculate the current with Ohm's Law is crucial for studying and designing circuits.
Resistor Power Dissipation
Resistor power dissipation refers to the process by which a resistor converts electrical energy into heat energy. The power dissipated by a resistor can be calculated with the formula:
  • \( P = I^2 R \)
This equation shows that the power dissipated is directly proportional to both the square of the current and the resistance.
For example, if you have a larger current or resistance, more electrical energy will be turned into heat.
In practice, understanding how resistors dissipate power is vital to ensure that they operate within safe limits and do not overheat, which can prevent potential circuit damage. Resistors in circuits must be appropriately rated to handle the power they dissipate.

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Most popular questions from this chapter

Some types of spiders build webs that consist of threads made of dry silk coated with a solution of a variety of compounds. This coating leaves the threads, which are used to capture prey, hygroscopic-that is, they attract water from the atmosphere. It has been hypothesized that this aqueous coating makes the threads good electrical conductors. To test the electrical properties of coated thread, researchers placed a \(5 \mathrm{~mm}\) length of thread between two electrical contacts. The researchers stretched the thread in \(1 \mathrm{~mm}\) increments to more than twice its original length, and then allowed it to return to its original length, again in \(1 \mathrm{~mm}\) increments. Some of the resistance measurements are given in the table: $$\begin{array}{l|cccccccc}\begin{array}{l}\text { Resistance of } \\\\\text { thread }\left(10^{9} \Omega\right)\end{array} & 9 & 19 & 41 & 63 & 102 & 76 & 50 & 24 \\ \hline \begin{array}{l}\text { Length of } \\\\\text { thread }(\mathrm{mm})\end{array} & 5 & 7 & 9 & 11 & 13 & 9 & 7 & 5\end{array}$$ In another experiment, a piece of the web is suspended so that it can move freely. When either a positively charged object or a negatively charged object is brought near the web, the thread is observed to move toward the charged object. What is the best interpretation of this observation? The web is A. a negatively charged conductor. B. a positively charged conductor. C. either a positively or negatively charged conductor. D. an electrically neutral conductor.

You have two \(10 \Omega\) resistors and one \(40 \Omega\) resistor. What combination of these will produce an equivalent resistance of (a) \(45 \Omega\), (b) \(18 \Omega,(\mathrm{c}) 4.44 \Omega ?\) 19.6 Kirchhoff's Rules

A gold wire \(6.40 \mathrm{~m}\) long and of diameter \(0.840 \mathrm{~mm}\) carries a current of 1.15 A. Find (a) the resistance of this wire and (b) the potential difference between its ends.

What is the resistance of a Nichrome wire at \(0.0^{\circ} \mathrm{C}\) if its resistance is \(100.00 \Omega\) at \(11.5^{\circ} \mathrm{C}\) ? The temperature coefficient of resistivity for Nichrome is \(0.00040\left(\mathrm{C}^{\circ}\right)^{-1}\)

An \(1800 \mathrm{~W}\) toaster, a \(1400 \mathrm{~W}\) electric frying pan, and a 75 W lamp are plugged into the same electrical outlet in a \(20 \mathrm{~A}, 120 \mathrm{~V}\) circuit. (Note: When plugged into the same outlet, the three devices are in parallel with each other across the \(120 \mathrm{~V}\) outlet. \()\) (a) What current is drawn by each device? (b) Will this combination blow the circuit breaker?
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