What happens to voltage across a capacitor?
2 Answers
The voltage across a capacitor changes depending on whether it's charging or discharging.
1. **Charging**: When a voltage is applied to a capacitor, it starts to store energy by accumulating charge on its plates. The voltage across the capacitor increases gradually and approaches the supply voltage. The relationship is typically exponential, described by the equation:
\[
V(t) = V_0 (1 - e^{-\frac{t}{RC}})
\]
where \( V_0 \) is the supply voltage, \( R \) is the resistance in the circuit, \( C \) is the capacitance, and \( t \) is time.
2. **Discharging**: When the capacitor is disconnected from the power source and allowed to discharge through a resistor, the voltage decreases exponentially:
\[
V(t) = V_0 e^{-\frac{t}{RC}}
\]
Here, \( V_0 \) is the initial voltage across the capacitor at the moment of disconnection.
In both cases, the time constant \( \tau = RC \) determines how quickly the voltage changes. A larger time constant means a slower change in voltage.
1. **Charging**: When a voltage is applied to a capacitor, it starts to store energy by accumulating charge on its plates. The voltage across the capacitor increases gradually and approaches the supply voltage. The relationship is typically exponential, described by the equation:
\[
V(t) = V_0 (1 - e^{-\frac{t}{RC}})
\]
where \( V_0 \) is the supply voltage, \( R \) is the resistance in the circuit, \( C \) is the capacitance, and \( t \) is time.
2. **Discharging**: When the capacitor is disconnected from the power source and allowed to discharge through a resistor, the voltage decreases exponentially:
\[
V(t) = V_0 e^{-\frac{t}{RC}}
\]
Here, \( V_0 \) is the initial voltage across the capacitor at the moment of disconnection.
In both cases, the time constant \( \tau = RC \) determines how quickly the voltage changes. A larger time constant means a slower change in voltage.
The behavior of voltage across a capacitor is influenced by how the capacitor is charged or discharged, and it's central to many electronic circuits and systems. Here’s a detailed explanation:
### **1. Charging a Capacitor**
When a capacitor is connected to a voltage source, such as a battery, it starts to charge. The process can be described by the following stages:
- **Initial State**: When you first connect the capacitor to a voltage source, there is no charge on the capacitor, and the voltage across it is initially zero.
- **Charging Phase**: As current flows from the voltage source into the capacitor, the charge on the capacitor increases. The voltage across the capacitor rises gradually as the capacitor accumulates more charge.
- **Voltage Increase**: The voltage \( V_C \) across the capacitor as it charges can be described by the equation:
\[
V_C(t) = V_0 \left(1 - e^{-\frac{t}{RC}}\right)
\]
where:
- \( V_0 \) is the voltage of the source.
- \( R \) is the resistance in the circuit.
- \( C \) is the capacitance.
- \( t \) is the time elapsed since charging began.
In this equation, \( RC \) is known as the time constant (\( \tau \)). The time constant determines how quickly the capacitor charges. After a time period equal to \( \tau \), the capacitor will have charged to about 63.2% of the supply voltage.
- **Fully Charged State**: Eventually, the capacitor becomes fully charged, meaning the voltage across it equals the supply voltage. At this point, the current flow stops, and the voltage across the capacitor remains constant as long as the voltage source is connected.
### **2. Discharging a Capacitor**
When the capacitor is disconnected from the charging source and connected to a load or a different circuit, it will start to discharge. The voltage across the capacitor during this process follows a different pattern:
- **Initial State**: At the start of discharge, the voltage across the capacitor is equal to the voltage it had when fully charged.
- **Discharge Phase**: As the capacitor discharges, the charge decreases, and so does the voltage across it. The voltage \( V_C(t) \) during discharge can be described by:
\[
V_C(t) = V_0 e^{-\frac{t}{RC}}
\]
where:
- \( V_0 \) is the initial voltage across the capacitor.
- \( R \) and \( C \) are as before.
- \( t \) is the time elapsed since the start of discharge.
Here, the voltage decreases exponentially with time, approaching zero as time goes on.
- **Fully Discharged State**: Over a long period, the voltage across the capacitor eventually becomes negligible. However, it never becomes exactly zero in a practical sense.
### **3. Capacitor in AC Circuits**
In alternating current (AC) circuits, the voltage across a capacitor varies continuously with the AC signal. The capacitor’s response to AC voltage involves complex interactions:
- **Impedance**: The opposition a capacitor presents to AC voltage is called capacitive reactance (\( X_C \)), and it decreases with increasing frequency. This relationship is given by:
\[
X_C = \frac{1}{2 \pi f C}
\]
where \( f \) is the frequency of the AC signal.
- **Phase Shift**: The voltage across a capacitor lags the current by 90 degrees (or a quarter of a cycle) in an AC circuit. This phase shift affects how capacitors are used in applications such as filtering, tuning, and phase correction.
### **Summary**
- **Charging**: Voltage across a capacitor increases exponentially until it reaches the supply voltage.
- **Discharging**: Voltage across a capacitor decreases exponentially until it approaches zero.
- **AC Circuits**: Voltage across a capacitor varies with the AC signal, and the capacitor's impedance decreases with increasing frequency.
Understanding these behaviors is crucial for designing and analyzing circuits involving capacitors, whether for filtering, timing, energy storage, or signal processing applications.
### **1. Charging a Capacitor**
When a capacitor is connected to a voltage source, such as a battery, it starts to charge. The process can be described by the following stages:
- **Initial State**: When you first connect the capacitor to a voltage source, there is no charge on the capacitor, and the voltage across it is initially zero.
- **Charging Phase**: As current flows from the voltage source into the capacitor, the charge on the capacitor increases. The voltage across the capacitor rises gradually as the capacitor accumulates more charge.
- **Voltage Increase**: The voltage \( V_C \) across the capacitor as it charges can be described by the equation:
\[
V_C(t) = V_0 \left(1 - e^{-\frac{t}{RC}}\right)
\]
where:
- \( V_0 \) is the voltage of the source.
- \( R \) is the resistance in the circuit.
- \( C \) is the capacitance.
- \( t \) is the time elapsed since charging began.
In this equation, \( RC \) is known as the time constant (\( \tau \)). The time constant determines how quickly the capacitor charges. After a time period equal to \( \tau \), the capacitor will have charged to about 63.2% of the supply voltage.
- **Fully Charged State**: Eventually, the capacitor becomes fully charged, meaning the voltage across it equals the supply voltage. At this point, the current flow stops, and the voltage across the capacitor remains constant as long as the voltage source is connected.
### **2. Discharging a Capacitor**
When the capacitor is disconnected from the charging source and connected to a load or a different circuit, it will start to discharge. The voltage across the capacitor during this process follows a different pattern:
- **Initial State**: At the start of discharge, the voltage across the capacitor is equal to the voltage it had when fully charged.
- **Discharge Phase**: As the capacitor discharges, the charge decreases, and so does the voltage across it. The voltage \( V_C(t) \) during discharge can be described by:
\[
V_C(t) = V_0 e^{-\frac{t}{RC}}
\]
where:
- \( V_0 \) is the initial voltage across the capacitor.
- \( R \) and \( C \) are as before.
- \( t \) is the time elapsed since the start of discharge.
Here, the voltage decreases exponentially with time, approaching zero as time goes on.
- **Fully Discharged State**: Over a long period, the voltage across the capacitor eventually becomes negligible. However, it never becomes exactly zero in a practical sense.
### **3. Capacitor in AC Circuits**
In alternating current (AC) circuits, the voltage across a capacitor varies continuously with the AC signal. The capacitor’s response to AC voltage involves complex interactions:
- **Impedance**: The opposition a capacitor presents to AC voltage is called capacitive reactance (\( X_C \)), and it decreases with increasing frequency. This relationship is given by:
\[
X_C = \frac{1}{2 \pi f C}
\]
where \( f \) is the frequency of the AC signal.
- **Phase Shift**: The voltage across a capacitor lags the current by 90 degrees (or a quarter of a cycle) in an AC circuit. This phase shift affects how capacitors are used in applications such as filtering, tuning, and phase correction.
### **Summary**
- **Charging**: Voltage across a capacitor increases exponentially until it reaches the supply voltage.
- **Discharging**: Voltage across a capacitor decreases exponentially until it approaches zero.
- **AC Circuits**: Voltage across a capacitor varies with the AC signal, and the capacitor's impedance decreases with increasing frequency.
Understanding these behaviors is crucial for designing and analyzing circuits involving capacitors, whether for filtering, timing, energy storage, or signal processing applications.