Skip to main content
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
All textbooksGiancoli Douglas 5th editionCh. 26 - DC CircuitsProblem 34
Chapter 25, Problem 34

(II) Determine the magnitudes and directions of the currents in each resistor shown in Fig. 26–57. The batteries have emfs of ε1 = 9.0V and ε2 = 12.0V and the resistors have values of R1 = 25 Ω, R2 = 48 Ω, and R3 = 35 Ω.
(a) Ignore internal resistance of the batteries.
(b) Assume each battery has internal resistance r = 1.0 Ω.

Verified step by step guidance
1
Step 1: Identify the circuit configuration and label the currents. Analyze the circuit to determine if it is a series, parallel, or a combination circuit. Assign current variables (e.g., I₁, I₂, I₃) to each branch of the circuit and define their directions (assume directions initially; corrections can be made later if needed).
Step 2: Apply Kirchhoff's Voltage Law (KVL) to each loop in the circuit. For each loop, write an equation summing the voltage drops across resistors and batteries, ensuring to account for the polarity of the batteries and the assumed current directions. For example, for a loop containing a resistor R and a battery ε, the voltage drop is given by V = IR, and the battery contributes ε (positive or negative depending on direction).
Step 3: Apply Kirchhoff's Current Law (KCL) at any junctions in the circuit. This law states that the sum of currents entering a junction equals the sum of currents leaving the junction. Write equations based on this principle to relate the currents in different branches.
Step 4: Solve the system of equations obtained from KVL and KCL. Use algebraic methods (e.g., substitution or matrix methods) to solve for the unknown currents (I₁, I₂, I₃). Ensure to include the internal resistance of the batteries (r = 1.0 Ω) in part (b) by adding the term rI to the voltage drop across each battery.
Step 5: Determine the magnitudes and directions of the currents. If any current value is negative, it means the actual direction of the current is opposite to the initially assumed direction. Interpret the results and verify that they satisfy all the equations and physical constraints of the circuit.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
10m
Was this helpful?

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 allows us to calculate the current through each resistor when the voltage and resistance values are known.

Kirchhoff's Laws

Kirchhoff's Laws consist of two principles that govern the behavior of electrical circuits. Kirchhoff's Current Law (KCL) states that the total current entering a junction must equal the total current leaving the junction, while Kirchhoff's Voltage Law (KVL) states that the sum of the electrical potential differences (voltage) around any closed circuit loop must equal zero. These laws are fundamental for analyzing complex circuits with multiple components.

Series and Parallel Circuits

In electrical circuits, components can be arranged in series or parallel configurations. In a series circuit, the same current flows through all components, and the total resistance is the sum of individual resistances. In a parallel circuit, the voltage across each component is the same, and the total current is the sum of the currents through each branch. Understanding these configurations is crucial for determining how current flows and how to calculate equivalent resistances in the given circuit.
Related Practice
Textbook Question

(III) Determine the net resistance in Fig. 26–61 (a) between points a and c, and (b) between points a and b. Assume R' ≠ R. [Hint: Apply an emf between the two points in each case and determine currents; use symmetry at junctions.]

1
views
Textbook Question

(III) (a) Determine the currents I1, I2, and I3 in Fig. 26–58. Assume the internal resistance of each battery is r = 1.0 Ω.


(b) What is the terminal voltage of the 6.0-V battery?

1
views
Textbook Question

A voltage V is applied to n identical resistors connected in parallel. If the resistors are instead all connected in series with the applied voltage, show that the power transformed is decreased by a factor n².

1
views
Textbook Question

(III) If the 25-Ω resistor in Fig. 26–59 is shorted out (resistance = 0 ), what then would be the current through the 15-Ω resistor?

2
views
Textbook Question

[In these Problems neglect the internal resistance of a battery unless the Problem refers to it.]


(III) You are designing a wire resistance heater to heat an enclosed container of gas. For the apparatus to function properly, this heater must transfer heat to the gas at a very constant rate. While in operation, the resistance of the heater will always be close to the value R = R₀, but may fluctuate slightly causing its resistance to vary a small amount ∆R ( << R₀ ). To maintain the heater at constant power, you design the circuit shown in Fig. 26–50, which includes two resistors, each of resistance R′. Determine the value for R′ so that the heater power P will remain constant even if its resistance R fluctuates by a small amount. [Hint: If ∆R << R₀ , then ΔP≈ΔRdPdR∣R=R0\(\Delta\) P\(\approx\) \(\Delta\) R\(\left\). \(\frac{dP}{dR}\]\right\)|_{R=R_{0}}]

1
views
Textbook Question

(II) (a) What is the potential difference between points a and d in Fig. 26–55 (similar to Fig. 26–12, Example 26–8), and (b) what is the terminal voltage of each battery?

1
views