College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Question
![**Finding the Unknown Resistor (R) in a Circuit**
In the given circuit diagram, we have:
- Two voltage sources in series: one with 6 V and another with 12 V, contributing to a total voltage of 18 V across the entire circuit.
- A parallel combination of three resistors:
- One resistor of 2 ohms (Ω)
- Two resistors of 4 ohms (Ω) each
The total current flowing through the circuit is given as \( I = 2 \, \text{A} \).
**Steps to Find the Unknown Resistor (R):**
1. **Find the Total Resistance of the Parallel Section:**
- The parallel resistors are:
- 2 Ω and two 4 Ω in parallel.
- The formula to find the total resistance \( R_p \) in parallel is:
\[
\frac{1}{R_p} = \frac{1}{2} + \frac{1}{4} + \frac{1}{4}
\]
- This simplifies to:
\[
\frac{1}{R_p} = \frac{1}{2} + \frac{1}{2} = 1
\]
- Thus, \( R_p = 1 \, \Omega \).
2. **Calculate the Total Resistance in the Circuit:**
- With \( R_p = 1 \, \Omega \), and assuming the unknown resistor \( R \) is in series with \( R_p \), the equation becomes:
\[
R_\text{total} = R_p + R = 1 + R
\]
3. **Determine the Unknown Resistor (R):**
- Using Ohm's law, \( V = I \times R \), the total voltage is 18 V and the total current is 2 A, so:
\[
18 = 2 \times (1 + R)
\]
- Solving for \( R \):
\[
18 = 2 + 2R \quad \Rightarrow \quad 16 = 2R \quad \Rightarrow \quad R = 8 \, \Omega
\]
**Conclusion:**
The unknown resistor \( R \) in the circuit is 8 ohms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2f5b0ad-df1d-48c6-a9a3-5338b8167143%2Fc713388d-70d0-4852-81b0-4565209a2422%2F2664xji_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Unknown Resistor (R) in a Circuit**
In the given circuit diagram, we have:
- Two voltage sources in series: one with 6 V and another with 12 V, contributing to a total voltage of 18 V across the entire circuit.
- A parallel combination of three resistors:
- One resistor of 2 ohms (Ω)
- Two resistors of 4 ohms (Ω) each
The total current flowing through the circuit is given as \( I = 2 \, \text{A} \).
**Steps to Find the Unknown Resistor (R):**
1. **Find the Total Resistance of the Parallel Section:**
- The parallel resistors are:
- 2 Ω and two 4 Ω in parallel.
- The formula to find the total resistance \( R_p \) in parallel is:
\[
\frac{1}{R_p} = \frac{1}{2} + \frac{1}{4} + \frac{1}{4}
\]
- This simplifies to:
\[
\frac{1}{R_p} = \frac{1}{2} + \frac{1}{2} = 1
\]
- Thus, \( R_p = 1 \, \Omega \).
2. **Calculate the Total Resistance in the Circuit:**
- With \( R_p = 1 \, \Omega \), and assuming the unknown resistor \( R \) is in series with \( R_p \), the equation becomes:
\[
R_\text{total} = R_p + R = 1 + R
\]
3. **Determine the Unknown Resistor (R):**
- Using Ohm's law, \( V = I \times R \), the total voltage is 18 V and the total current is 2 A, so:
\[
18 = 2 \times (1 + R)
\]
- Solving for \( R \):
\[
18 = 2 + 2R \quad \Rightarrow \quad 16 = 2R \quad \Rightarrow \quad R = 8 \, \Omega
\]
**Conclusion:**
The unknown resistor \( R \) in the circuit is 8 ohms.
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