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)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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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:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9addf70b-7b61-4f39-8ae4-a2b3648accb6%2Ff58cb02e-dcb8-433c-8bb6-dfa2caea404c%2Fcypf1e_processed.png&w=3840&q=75)
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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