Capacitors (6.1.1)
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Capacitance is a measure of an object’s ability to store charge. A capacitor is an electrical device that stores charge.
Capacitors generally consist of two parallel metal plates separated by an electrically insulating material such as air. When charged by a cell or battery in a circuit, the plates of a capacitor store equal and opposite charges.
Capacitance is defined by the equation:
Where in a:
- parallel plate capacitor is the magnitude of charge stored on each plate and is the potential difference between the plates.
- charged conductor is the charge of the conductor and is its electric potential.
The unit of capacitance is the farad
One farad is equal to the capacitance of a capacitor that stores a charge of 1 C on each plate when the potential difference across the plates is 1 V.
It is important to note that one farad is very large, so practical capacitance values are often given in microfarads nanofarads or picofarads )
Question walkthrough
Finding Voltage from Charge and Capacitance
Rearrange C = Q/V to find the potential difference across a capacitor given its capacitance and stored charge.
An uncharged capacitor will accumulate charge when connected to a battery or cell. To prevent dangerously high currents, capacitors will usually be charged in series with a resistor.
As electrons build up on one plate during charging, a corresponding shortage of electrons is left on the other plate.
When a capacitor is charging:
- voltage across the capacitor increases until it reaches the voltage of the battery or cell,
- the charge on the plates of the capacitor increases, and
- the current in the circuit decreases.
Discharging a capacitor occurs when a charged capacitor is connected in a closed circuit. When discharging, electrons flow back from the negative plate to the positive plate until the potential difference across the capacitor falls to zero.
When a capacitor is discharging:
- voltage across the capacitor decreases
- charge (on the plates on the capacitor) decreases
- current (in the circuit) decreases but is in the opposite direction to when charging.
In the circuit setup shown, the capacitor will charge when connected to A and discharge when connected to B.
For both charging and discharging capacitors:
- Throughout the charging/discharging process, the charge on each plate is equal and opposite.
- The magnitude of the current decreases exponentially (but in opposite directions) during charge and discharge.
- A resistor or resistive circuit is needed to prevent dangerously high currents.
The total capacitance of two or more capacitors in series is given by:
An example of total capacitance with three capacitors in series would be:
Recall the correct technique to find the total capacitance of two or more capacitors in series.
Question walkthrough
Charge and Capacitance for Series Capacitors
Calculate the total capacitance of series capacitors and the charge stored on each, given a supply voltage.
Question walkthrough
Deriving Total Capacitance for Series Capacitors
Prove that reciprocal capacitance adds for capacitors in series by combining conservation of charge with the sum of individual voltages.
The total capacitance of two or more capacitors in parallel is given by:
An example of calculating the total capacitance with three capacitors in parallel is:
Question walkthrough
Finding Charge on a Capacitor in Parallel
Calculate the total capacitance of parallel capacitors and use it to find the charge stored on one branch at a given voltage.
Question walkthrough
Deriving Total Capacitance for Parallel Capacitors
Prove that capacitance adds for capacitors in parallel by combining conservation of charge with the shared voltage across each branch.
Determining the capacitance of a capacitor without a datalogger:
- Set up the circuit shown below.
- Charge the capacitor by placing the switch at position A.
- The capacitor is fully charged when the reading on the voltmeter becomes constant. Record this constant value (it should be equal to the voltage of the cell).
- Close the switch and start a timer.
- Vary the resistance of the variable resistor such that the current is kept constant.
- After an amount of time , the current will become zero.
- Record the time and the (average) current during this time
- Use the equation to determine the total charge stored by the capacitor.
- Use the equation to determine the capacitance of the capacitor.
It is important to note that the current naturally decreases over time during capacitor charging. Varying the circuit’s resistance keeps the current approximately constant throughout the charging process.
Investigating capacitors in series and parallel using an ammeter and a voltmeter Set up the circuits shown below, or any other arrangement of series and parallel capacitors that you wish to investigate. Follow the procedure below for each circuit setup:
- Charge the capacitor by placing the switch at position A.
- The capacitor is fully charged when the reading on the voltmeter becomes constant.
- Record this constant value (it should be equal to the voltage of the cell).
- Close the switch and start a timer.
- Vary the resistance of the variable resistor so that the current is kept constant.
- After a certain amount of time, the current will become zero. Record the time, and the (average) current during this time, .
- Use the equation to determine the total charge stored by the capacitors.
- Use the equation to determine the total capacitance of the combination of capacitors.
It is important to note that the current will naturally decrease over time when charging a capacitor. Varying the circuit’s resistance keeps the current approximately constant throughout the charging process.