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f C = 25 µF, , R1 = 30 N, R2 = 17 Q, R3 = 10 0, and R4 = 20 N, determine the charge Q on capacitor C. (Q =91.25 µC)

f C = 25 µF, , R1 = 30 N, R2 = 17 Q, R3 = 10 0, and R4 = 20 N, determine the charge Q on capacitor C. (Q =91.25 µC)

College Physics
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
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**Title: Calculating the Total Charge Stored on Capacitors in Fully Charged RC Circuits** **Topic: RC Circuits and Capacitors** In each RC circuit shown below, \( V_{bat} = 12V \) and the switch has been closed for a **very long time** (circled in purple), allowing the capacitors in the circuits to become fully charged. **Determine the total charge stored on each capacitor.** The answers are provided in parenthesis. Furthermore, **t → ∞** is illustrated below with a purple arrow indicating the time has approached infinity, meaning the capacitors have reached their maximum charge. This text aims to explain the charging process of capacitors in RC circuits and provide the necessary steps to compute the total charge stored on each capacitor after a long period. 1. **Initial Conditions**: Note the given voltage (\( V_{bat} = 12V \)). 2. **Fully Charged Condition**: Recognize that the capacitors are fully charged after the switch has remained closed for a very long time. Let's break down the key principles and equations necessary for this calculation: --- **Key Principles and Equations** 1. **Capacitance Formula**: Q = CV - Where: - Q = Charge (in coulombs, C) - C = Capacitance (in farads, F) - V = Voltage across the capacitor (in volts, V) 2. **Understanding the fully charged condition**: At \( t → ∞ \), the capacitors will hold the maximum charge, with the battery providing a constant voltage \( V_{bat} \). 3. **Total Charge Calculation**: Use the formula \( Q = CV \) for each capacitor, using the given \( V_{bat} \). --- **Example Problem Statement and Solution Approach**: - With \( V_{bat} = 12V \), for a given capacitance \( C \), determine the total charge \( Q \) stored. - Apply the formula \( Q = CV \): - If \( C = 1F \): - \( Q = 1F \times 12V = 12C \) - If \( C = 100 \mu F \): - \( Q = 100 \times 10^{-6}F \times 12V = 1.2 \times 10^{-3}C =
Transcribed Image Text:**Title: Calculating the Total Charge Stored on Capacitors in Fully Charged RC Circuits** **Topic: RC Circuits and Capacitors** In each RC circuit shown below, \( V_{bat} = 12V \) and the switch has been closed for a **very long time** (circled in purple), allowing the capacitors in the circuits to become fully charged. **Determine the total charge stored on each capacitor.** The answers are provided in parenthesis. Furthermore, **t → ∞** is illustrated below with a purple arrow indicating the time has approached infinity, meaning the capacitors have reached their maximum charge. This text aims to explain the charging process of capacitors in RC circuits and provide the necessary steps to compute the total charge stored on each capacitor after a long period. 1. **Initial Conditions**: Note the given voltage (\( V_{bat} = 12V \)). 2. **Fully Charged Condition**: Recognize that the capacitors are fully charged after the switch has remained closed for a very long time. Let's break down the key principles and equations necessary for this calculation: --- **Key Principles and Equations** 1. **Capacitance Formula**: Q = CV - Where: - Q = Charge (in coulombs, C) - C = Capacitance (in farads, F) - V = Voltage across the capacitor (in volts, V) 2. **Understanding the fully charged condition**: At \( t → ∞ \), the capacitors will hold the maximum charge, with the battery providing a constant voltage \( V_{bat} \). 3. **Total Charge Calculation**: Use the formula \( Q = CV \) for each capacitor, using the given \( V_{bat} \). --- **Example Problem Statement and Solution Approach**: - With \( V_{bat} = 12V \), for a given capacitance \( C \), determine the total charge \( Q \) stored. - Apply the formula \( Q = CV \): - If \( C = 1F \): - \( Q = 1F \times 12V = 12C \) - If \( C = 100 \mu F \): - \( Q = 100 \times 10^{-6}F \times 12V = 1.2 \times 10^{-3}C =
### Problem 5: Calculating the Charge on a Capacitor in a Complex Circuit #### Given: - **Capacitance, \( C \) = 25 μF** - **Resistor \( R_1 \) = 30 Ω** - **Resistor \( R_2 \) = 17 Ω** - **Resistor \( R_3 \) = 10 Ω** - **Resistor \( R_4 \) = 20 Ω** - **Charge on the capacitor \( Q \) = 91.25 μC** #### Goal: Determine the charge \( Q \) on the capacitor \( C \). ##### Circuit Diagram: The circuit consists of a combination of resistors and a capacitor. Here's a detailed description: - A battery is connected in series with a switch. - \( R_1 \) and \( R_2 \) are arranged in series. - The above series combination is in parallel with another branch: - The parallel branch has \( R_3 \) and the capacitor \( C \) connected in series. - \( R_4 \) is connected in parallel to the combination of \( R_1 \) and \( R_2 \), and \( R_3 \) and \( C \). - The entire network is connected back to the battery. Note: The solution to finding \( Q \) involves understanding complex circuit analysis, which can be solved using techniques such as Kirchhoff's laws, Thevenin’s theorem, or the principle of superposition. The given charge, \( Q = 91.25 \text{ μC} \), simplifies the task but is usually derived through these methods. #### Explanation of the Calculation Process (General Approach): 1. **Identify Series and Parallel Components:** - Determine equivalent resistances. 2. **Calculate Equivalent Resistance:** - Use formulas for series and parallel resistances. 3. **Apply Kirchhoff's Laws:** - Set up and solve the equations based on voltage and current continuity. 4. **Determine Voltage Across Capacitor:** - Using the derived equivalent resistance, find the voltage across the capacitor's branch. 5. **Calculate Charge Using Capacitor Formula:** - Use \( Q = CV \) to find the charge. This problem outlines the fundamentals of circuit analysis and capacitor behavior in complex circuits. By mastering these principles, one can solve a range of electrical engineering problems.
Transcribed Image Text:### Problem 5: Calculating the Charge on a Capacitor in a Complex Circuit #### Given: - **Capacitance, \( C \) = 25 μF** - **Resistor \( R_1 \) = 30 Ω** - **Resistor \( R_2 \) = 17 Ω** - **Resistor \( R_3 \) = 10 Ω** - **Resistor \( R_4 \) = 20 Ω** - **Charge on the capacitor \( Q \) = 91.25 μC** #### Goal: Determine the charge \( Q \) on the capacitor \( C \). ##### Circuit Diagram: The circuit consists of a combination of resistors and a capacitor. Here's a detailed description: - A battery is connected in series with a switch. - \( R_1 \) and \( R_2 \) are arranged in series. - The above series combination is in parallel with another branch: - The parallel branch has \( R_3 \) and the capacitor \( C \) connected in series. - \( R_4 \) is connected in parallel to the combination of \( R_1 \) and \( R_2 \), and \( R_3 \) and \( C \). - The entire network is connected back to the battery. Note: The solution to finding \( Q \) involves understanding complex circuit analysis, which can be solved using techniques such as Kirchhoff's laws, Thevenin’s theorem, or the principle of superposition. The given charge, \( Q = 91.25 \text{ μC} \), simplifies the task but is usually derived through these methods. #### Explanation of the Calculation Process (General Approach): 1. **Identify Series and Parallel Components:** - Determine equivalent resistances. 2. **Calculate Equivalent Resistance:** - Use formulas for series and parallel resistances. 3. **Apply Kirchhoff's Laws:** - Set up and solve the equations based on voltage and current continuity. 4. **Determine Voltage Across Capacitor:** - Using the derived equivalent resistance, find the voltage across the capacitor's branch. 5. **Calculate Charge Using Capacitor Formula:** - Use \( Q = CV \) to find the charge. This problem outlines the fundamentals of circuit analysis and capacitor behavior in complex circuits. By mastering these principles, one can solve a range of electrical engineering problems.
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