Ch. 26 - DC Circuits

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 26 - DC Circuits

Problem 101

Chapter 25, Problem 101

Consider two unequal resistors, of resistance R₁ and R₂, that are connected either in series or in parallel. Fill in the Table below assuming the electric potential on the low-voltage end of the combination is V_A volts and the potential at the high-voltage end of the combination is V_B volts. First draw diagrams.

Verified step by step guidance

Step 1: Understand the problem setup. Two resistors, R₁ and R₂, are connected either in series or in parallel. The goal is to analyze the voltage distribution and equivalent resistance for each configuration. Begin by recalling the formulas for equivalent resistance in series and parallel connections.

Step 2: For the series configuration, the equivalent resistance is given by:

R_{eq} = R_{1} + R_{2}

. The voltage across each resistor can be calculated using Ohm's Law:

V_{1} = I R_{1}

and

V_{2} = I R_{2}

, where the current I is the same through both resistors.

Step 3: For the parallel configuration, the equivalent resistance is given by:

R_{eq} = \frac{1}{\frac{1}{R_{1}}}

. The voltage across each resistor is the same and equal to the total voltage:

V_{1} = V_{2} = V_{B} - V_{A}

. The current through each resistor can be calculated using Ohm's Law:

I_{1} = \frac{V_{B}}{R_{1}}

and

I_{2} = \frac{V_{B}}{R_{2}}

Step 4: Draw diagrams for both configurations. For the series connection, represent R₁ and R₂ connected end-to-end, with the current flowing through both resistors sequentially. For the parallel connection, represent R₁ and R₂ connected side-by-side, with the voltage across both resistors being the same.

Step 5: Fill in the table based on the derived formulas and diagrams. For each configuration, calculate the equivalent resistance, voltage across each resistor, and current through each resistor using the relationships established in Steps 2 and 3. Ensure the table reflects the differences between series and parallel configurations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed mathematically as V = IR. Understanding this law is essential for analyzing circuits, as it helps determine how voltage, current, and resistance interact in both series and parallel configurations.

Series and Parallel Circuits

In a series circuit, resistors are connected end-to-end, and the total resistance is the sum of the individual resistances (R_total = R₁ + R₂). In contrast, in a parallel circuit, resistors are connected across the same two points, and the total resistance can be calculated using the formula 1/R_total = 1/R₁ + 1/R₂. Understanding these configurations is crucial for predicting how voltage and current will behave in the circuit.

Voltage Division and Current Division

Voltage division refers to the way voltage is distributed across resistors in a series circuit, where the voltage drop across each resistor is proportional to its resistance. Conversely, current division describes how current is distributed among parallel resistors, where the current through each resistor is inversely proportional to its resistance. These principles are vital for analyzing the behavior of circuits and calculating the voltage and current at different points.

Related Practice

Textbook Question

Measurements made on circuits that contain large resistances can be confusing. Consider a circuit powered by a battery ε = 15.000 V with a 10.00-MΩ resistor in series with an unknown resistor R. As shown in Fig. 26–92, a particular voltmeter reads V₁ = 366 mV when connected across the 10.00 -MΩ resistor and this meter reads V₂ = 7.317 V when connected across R. Determine the value of R. [Hint: Define R_V as the voltmeter’s internal resistance.]

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Textbook Question

The circuit shown in Fig. 26–89 is a primitive 4-bit digital-to-analog converter (DAC). In this circuit, to represent each digit (2ⁿ) of a binary number, a “1” has the nᵗʰ switch closed whereas zero (“0”) has the switch open. For example, 0010 is represented by closing switch n = 1, while all other switches are open. Show that the voltage V across the 1.0 - Ω resistor for the binary numbers 0001, 0010, 0100, and 1010 (which represent 1, 2, 4, 10) follows the pattern that you expect for a 4-bit DAC.

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Textbook Question

The performance of the starter circuit in a car can be significantly degraded by a small amount of corrosion on a battery terminal. Figure 26–88a depicts a properly functioning circuit with a battery (12.5-V emf, 0.02-Ω internal resistance) attached via corrosion-free cables to a starter motor of resistance R_s = 0.15Ω. Sometime later, corrosion between a battery terminal and a starter cable introduces an extra series resistance of only R_C = 0.10Ω into the circuit as suggested in Fig. 26–88b. Let P₀ be the power delivered to the starter in the circuit free of corrosion, and let P be the power delivered to the starter with corrosion. Determine the ratio P/P₀.

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