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20K Ω M 200 V 50 V 5ΚΩ 6ΚΩ 20 μ F. The switch has been open for a long time. It is closed at t = 0. Find the voltage across the capacitor v (t), for t≥ 0.

20K Ω M 200 V 50 V 5ΚΩ 6ΚΩ 20 μ F. The switch has been open for a long time. It is closed at t = 0. Find the voltage across the capacitor v (t), for t≥ 0.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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### Transcription for Educational Website #### Circuit Diagram Analysis The given image depicts an electrical circuit with the following components: - Two voltage sources: - 200 V connected in series with a 20kΩ resistor. - 50 V connected in parallel with a 5kΩ resistor. - A switch that has been open for a long time and is closed at \( t = 0 \). - The switch then connects to a 6kΩ resistor in series with a 20μF capacitor. #### Circuit Explanation: - **Before \( t = 0 \) (When the switch is open):** - The 200 V source is connected to the 20kΩ resistor, and the 50 V source is connected to the 5kΩ resistor. - Since the switch is open, the 6kΩ resistor and the 20μF capacitor do not have any current flowing through them. - **At \( t = 0 \) (When the switch is closed):** - The switch is closed, connecting the 200 V source in series with the combination of the 6kΩ resistor and the 20μF capacitor in parallel with the 5kΩ resistor. #### Problem Statement: The task is to find the voltage across the capacitor \( v_c(t) \) for \( t \geq 0 \). #### Analysis Approach: 1. **Initial Conditions:** - Since the switch has been open for a long time, the capacitor can be considered initially uncharged. 2. **Closing the Switch:** - At \( t = 0 \), the capacitor will start charging up through the 6kΩ resistor. The voltage across the capacitor \( v_c(t) \) can be found using the differential equation governing the charging of a capacitor in an RC circuit. 3. **Formula:** - For an RC circuit, the voltage across the capacitor as a function of time is given by: \[ v_c(t) = V_s \left(1 - e^{-\frac{t}{RC}}\right) \] - Where: \[ R = 6kΩ \] \[ C = 20μF \] \[ V_s = 200V - V_{drop\ across\ 5kΩ} \] 4. **Time Constant:
Transcribed Image Text:### Transcription for Educational Website #### Circuit Diagram Analysis The given image depicts an electrical circuit with the following components: - Two voltage sources: - 200 V connected in series with a 20kΩ resistor. - 50 V connected in parallel with a 5kΩ resistor. - A switch that has been open for a long time and is closed at \( t = 0 \). - The switch then connects to a 6kΩ resistor in series with a 20μF capacitor. #### Circuit Explanation: - **Before \( t = 0 \) (When the switch is open):** - The 200 V source is connected to the 20kΩ resistor, and the 50 V source is connected to the 5kΩ resistor. - Since the switch is open, the 6kΩ resistor and the 20μF capacitor do not have any current flowing through them. - **At \( t = 0 \) (When the switch is closed):** - The switch is closed, connecting the 200 V source in series with the combination of the 6kΩ resistor and the 20μF capacitor in parallel with the 5kΩ resistor. #### Problem Statement: The task is to find the voltage across the capacitor \( v_c(t) \) for \( t \geq 0 \). #### Analysis Approach: 1. **Initial Conditions:** - Since the switch has been open for a long time, the capacitor can be considered initially uncharged. 2. **Closing the Switch:** - At \( t = 0 \), the capacitor will start charging up through the 6kΩ resistor. The voltage across the capacitor \( v_c(t) \) can be found using the differential equation governing the charging of a capacitor in an RC circuit. 3. **Formula:** - For an RC circuit, the voltage across the capacitor as a function of time is given by: \[ v_c(t) = V_s \left(1 - e^{-\frac{t}{RC}}\right) \] - Where: \[ R = 6kΩ \] \[ C = 20μF \] \[ V_s = 200V - V_{drop\ across\ 5kΩ} \] 4. **Time Constant:
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