Charging and discharging capacitors (6.1.3)
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A capacitor charges when connected to a power supply.
The current flows around the circuit, and charges build up on the capacitor plates, creating a potential difference.
A resistor is generally connected in series to prevent dangerously high currents from flowing.
Discharging occurs when the capacitor is disconnected from the power supply and connected to a resistor.
The stored energy is released, causing the voltage and current to decrease over time.
For a charging capacitor, the charge stored on the plates has a logarithmic growth rate, meaning the rate of increase slows with time.
The rate of increase (gradient of the against graph) tends to zero.
For a discharging capacitor, the charge stored on the plates decreases exponentially over time.
The rate of decrease (gradient of the against graph) tends to zero.
For a charging capacitor, the magnitude of current in the circuit decreases exponentially.
The rate of decrease (gradient of the against graph) tends to zero.
For a discharging capacitor, the current flows in the opposite direction. At the moment of discharge, the current is the negative of the initial current for the charging capacitor.
The magnitude of the current in a circuit with a discharging capacitor decreases exponentially.
The rate of decrease (gradient of the against graph) tends to zero.
It is important to note that the direction may be reversed (positive for discharging a capacitor and negative for charging a capacitor) depending on how the ammeter is connected relative to the circuit.
Investigating the charging of a capacitor.
- Connect the uncharged capacitor in series with a resistor, a power supply, an ammeter, and a switch, as shown in the image.
- Place a voltmeter in parallel with the capacitor.
- Close the switch to connect the power supply and simultaneously start a timer.
- Record values of voltage and current at regular intervals.
- Stop recording once the voltage across the capacitor reaches a steady state.
- Plot graphs of voltage against time and current against time.
- Alternatively, plot graphs of against and against . The gradient of these graphs will be
It is useful to know that instead of recording values manually, the ammeter and voltmeter can be connected to a datalogger. Once connected to a computer, more accurate graphs can be plotted, with readings taken at much smaller time intervals than is possible manually.
Investigating the discharging of a capacitor.
- Connect the fully charged capacitor in series with a resistor, an ammeter, and a switch, as shown in the image.
- Place a voltmeter in parallel with the capacitor.
- Close the switch to discharge the capacitor and simultaneously start a timer.
- Record values of voltage and current at regular intervals.
- Stop recording once the voltage across the capacitor reaches zero.
- Plot graphs of voltage against time and current against time.
- Alternativity plot graphs of against and against . The gradient of these graphs will be
Again, the ammeter and voltmeter can be connected to a datalogger, allowing more accurate graphs can be plotted.
Question walkthrough
Identifying the Discharge Current Graph
Identifies which graph correctly shows how ammeter current varies with time once a fully charged capacitor begins discharging through the switch.
The time constant of a capacitor–resistor circuit represents the time it takes for the voltage or charge on a discharging capacitor (or current in the circuit) to fall to times its initial value.
The time constant reflects how quickly energy stored in the capacitor is released or replenished.
For a charging capacitor, the voltage or charge (or current in the circuit) will be equal to times its initial value after one time constant \tau has passed.
It is important to note that mathematically, the time constant of a capacitor–resistor (measured in seconds) is given by:
Where:
- is the capacitance of the capacitor (measured in Farads), and
- is the resistance of the circuit (measured in Ohms).
It is important to note that a capacitor is considered to be fully charged or discharged after five time constants have passed. At this time, the capacitor will be over 99% charged or discharged.
Question walkthrough
Time Constant of a Series Circuit
Combines series resistors and series capacitors to find total resistance and capacitance, then calculates the time constant of the resulting RC circuit.
Question walkthrough
Finding Resistance from a Charging Graph
Uses the one-time-constant charge value read from a charging capacitor's Q-t graph to calculate the resistance of the resistor in the circuit.
The magnitude of the voltage charge and current for a charging capacitor as a function of time are given by:
Where:
- is the voltage of the power supply, and the final voltage across the capacitor,
- is the final charge across the capacitor,
- is the initial current in the circuit,
- is the resistance of the circuit,
- and, is the capacitance of the capacitor.
For a discharging capacitor discharging in series with a resistor, we have:
Where:
- is voltage, charge, or current, and
- is the initial voltage, charge, or current.
Taking the natural logarithm of both sides gives a linear relationship:
A graph of plotted against will have a gradient of and a -intercept of
The gradients and -intercepts are the same because they stem from the same fundamental equation governing exponential growth and decay.
For a charging capacitor charging in series with a resistor we have:
Where:
- is the voltage or charge, and
- is the initial voltage or charge.
Rearranging for and taking the natural logarithm of both sides gives a linear relationship:
A graph of plotted against will have a gradient of and a -intercept of
Question walkthrough
Finding Capacitance from Discharge Time
Rearranges the exponential discharge equation to calculate the capacitance of a capacitor given the resistance and time taken to fall to one-third of its initial voltage.
Question walkthrough
Current and Time Constant from Graph
Extracts the initial current and time constant of a charging capacitor circuit from the gradient and intercept of a linearised ln(I) against t graph.
Question walkthrough
Deriving the One Time-Constant Voltage
Uses the exponential discharge equation to show that capacitor voltage falls to approximately 37% of its initial value after one time constant, RC.
For a capacitor discharging across a circuit with a resistance we have:
Where:
- is the rate of flow of charge off of the capacitor,
- is the charge on the capacitor, is the resistance of the resistor, and
- is the capacitance of the capacitor.
It is important to note that as the charge decreases in a discharging capacitor, the change in charge is negative. Its magnitude equals the current in the circuit:
The voltage across the capacitor at time as it discharges is the same as the voltage across the resistor (as the sum of the potentials in a closed loop is zero), so the current in the circuit is also given by:
For the capacitor, we also have:
Combining these equations gives:
An example of how a capacitor discharges over time can be explored using iterative modelling with a spreadsheet.
- Start with the initial charge and time constant of the circuit.
- Choose a small-time interval compared to to improve accuracy.
- For each step, calculate the charge lost and subtract it from the previous charge to find the remaining charge.
- Repeat this process for subsequent intervals to model the full discharge.
Computer software, such as Excel or Google Sheets, can automate this process and plot charge versus time.
For a discharging capacitor charging in series with a resistor we have:
Where:
- is voltage, charge or current,
- and, is the initial voltage, charge or current.
The voltage–time graph for an exponentially decaying capacitor against time shows a rapid drop initially, followed by a slower decrease over time.
The exponential decay graph has the characteristic constant-ratio property: the charge (or current/voltage) decreases by the same fraction for equal time intervals.
It is important to note that this property applies to any fixed ratio of the quantity being measured over equal time intervals:
- For the ratio the corresponding time interval is the time constant .
- For the ratio the corresponding ratio is known as the half-life.
Question walkthrough
Capacitor Charge After Repeated Discharging
Applies repeated fractional decay to find a capacitor's charge 5 seconds after discharge begins, given the charge remaining after 1 second.