Capacitance (3.7.4)
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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.
Capacitors are used in various devices where rapid energy release is needed because of their ability to quickly discharge stored energy. Some common applications include:
There are advantages and disadvantages to using capacitors instead of batteries in various practical applications.
The diagram below shows a parallel plate capacitor, where the space between the two identical plates is a vacuum:
A battery connected to the capacitor creates a potential difference across the two plates, which forces electrons towards one plate and away from the other. The difference in the number of charge carriers between the plates results in two oppositely charged plates, creating an electric field between them.
The strength of the field between the two capacitor plates is dependent on the amount of charge stored on the plates: the more charge, the greater the electric field strength. The electric field strength is also dependent on the distance between the two plates: the shorter the distance, the greater the electric field strength.
It is important to note that capacitance is the amount of charge a component can store at a given potential difference defined by:
Where, in the context of a parallel plate capacitor:
- is the magnitude of charge stored on each plate, and
- is the potential difference between the plates.
The capacitance of a parallel plate capacitor in a vacuum is directly proportional to the area of overlap, since more charge can be stored in a larger area.
The capacitance of a parallel plate capacitor in a vacuum is also inversely proportional to the plate separation, since a smaller plate separation leads to a larger electric field between the plates, meaning a smaller voltage is required to store a given charge.
Therefore, the equation for the capacitance of a parallel plate capacitor is given by:
Where:
- is the capacitance of the capacitor,
- is the area of overlap,
- is the plate separation distance, and
- the proportionality constant is the permittivity of free space, .
When an insulator, also known as a dielectric, is inserted into the space between the plates of a capacitor, the equation for capacitance now incorporates the permittivity of the insulator, .
Where:
- is the relative permittivity,
- is the permittivity of free space.
Therefore, the capacitance of a parallel plate capacitor with a dielectric becomes:
Where:
- is the capacitance of the capacitor,
- is the area of overlap,
- is the plate separation distance,
Different dielectric materials have differing values for the relative permittivity, . For example, a vacuum has a relative permittivity of 1 by definition, and air has a value of 1.0006. The relative permittivity is a dimensionless quantity.
The table below gives the values of the relative permittivity for some common dielectric materials:
The permittivity of a material quantifies its response to an electric field. For the same applied voltage, a material with a higher permittivity has a stronger electric field. Therefore, a smaller voltage is required to store a given charge and this leads to a larger capacitance.
Question walkthrough
Calculating Capacitance of a Dielectric-Filled Capacitor
Calculate the capacitance of a parallel-plate capacitor filled with a perspex dielectric, given the plate dimensions, separation, and relative permittivity.
The potential difference across a capacitor is directly proportional to the charge it stores.
It is important to note that while a capacitor is charging, the charge stored on the plates increases linearly with the voltage across the plates; therefore, a graph of charge against voltage will be a straight line passing through the origin.
The gradient of the charge against voltage graph is equal to the capacitance of the capacitor:
The energy stored by a capacitor ( is equal to the area under the charge–voltage graph for the capacitor.
The area under the graph is a triangle, so it is given by:
For a capacitor with a voltage across its plates storing a charge we have:
It is useful to know that generally, energy is given by:
the integral of the charge with respect to the potential difference. You will not need to know this calculus for your physics exam, but it could be helpful in your understanding or for future learning.
It is important to note that the correct equation for the energy stored by a capacitor. Many students make this mistake in their exams, losing marks as a result.
Question walkthrough
Finding Energy and Capacitance from a V-Q Graph
Use the area under a voltage-charge graph to find energy stored, then find capacitance from the gradient.
The energy stored by a capacitor is given by:
Where:
- is the charge stored by the capacitor, and
- is the voltage across the capacitor.
Substituting where is the capacitance of the capacitor gives:
Substituting gives:
All three equations are equivalent and can be used to calculate the stored energy, depending on which two quantities (charge , capacitance or voltage ) are known.
Question walkthrough
Finding Voltage from Capacitor Energy
Rearrange the capacitor energy formula to find the potential difference given the stored energy and capacitance.
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.
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.
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.
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.
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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:
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.