Capacitance: Definition, Formula & Units

Just like batteries allow for portable energy storage, capacitors allow for temporary energy storage and are critical components of many circuits.

They allow large amounts of charges to be separated from each other and released in a sudden burst of energy, for use in such devices as flash cameras, as well as to mediate other electronics processes like converting between AC and DC power sources or charging and discharging magnetic fields, which is useful in tuning radio stations.

Capacitance is a measure of a non-conducting material's ability to store energy by creating a separation of charge across a potential difference (voltage). The material must be non-conducting, like glass or a PVC pipe, because otherwise the charges would flow through it, unable to stay separated.

Mathematically, an object's capacitance ​C​ is equal to the ratio of charge ​Q​ to voltage ​V​.

\(C=\frac{Q}{V}\)

The SI unit of capacitance is the ​farad​ (F); of charge, the ​coulomb​ (C); and of voltage, ​volts​ (V)​.​ The farad, named after electromagnetism pioneer Michael Faraday, is defined such that 1 farad is equal to 1 columb per volt, or 1 F = 1 C/V.

Any part of a circuit that separates charge in this way is called a ​capacitor​. Thus, following the equation above, any given capacitance of a capacitor ​C​ connected to a battery with a potential difference ​V​, will store electric charge ​Q​.

One common type of capacitor is a ​parallel plate capacitor​. In such a device, two plates of conduction material (like a metal) are held, as the name suggests, parallel to one another across some distance. In between the plates is a ​dielectric material​, also called an ​insulating material​.

This is something that does not allow charges to flow through it and thus can become polarized – the charges inside it reorient so all the positives are together on one side and all the negatives on the other – in the presence of an electric field.

Anyone can create a simple parallel plate capacitor using two sheets of metal foil as the plates and several sheets of paper as the insulator sandwiched between them.

The capacitance of a parallel plate capacitor depends on the area of one plates, or ​A​; the separation between them ​d​; and the dielectric constant ​κ​ of the material between them in this way:

\(C = \dfrac{κε_0A}{d}\)

The term ε₀ ("epsilon-naught") is the ​permittivity​ of free space, which is a constant equal to 8.854 × 10^-12 farads per meter (F/m). The dielectric constant ​κ​ is a unit-less quantity that can be looked up in a table, such as the one linked to this article.

Not all types of capacitors require parallel plates. Some are cylindrical, like a coaxial cable, or spherical, like a cell membrane (which ends up holding a charge by pumping positive potassium ions out of the cell and negative chloride ions into it).

A coaxial cable is widely used to deliver video, audio and communications data. Its cylindrical design consists of several layers of insulating dielectric materials between strong conducting sheets, often copper, all rolled up like a jelly roll.

This allows the cable to carry even weak electrical signals without degradation over long distances. Additionally, because the insulating and conducting layers are rolled up, a coaxial cable is able to provide this energy storage in a relatively small space – certainly in a smaller volume than parallel plate capacitors can.

One common application of capacitors is in an RC circuit, so named because it contains a resistor and a capacitor. Suppose two circuit components are connected in parallel, with a switch allowing the circuit to connect in one of two possible single loops: voltage source plus capacitor, or capacitor plus resistor.

When the capacitor is connected to the voltage source, current flows in the circuit, and it begins to build up a stored charge. When the switch is flipped and the capacitor is connected to the resistor, it will discharge and heat up the resistor.

The voltage, or potential difference, across the capacitor when it is charging is:

\(V_{capacitor} = V_{source}(1-e^{t/RC})\)

Where both ​_V_capacitor​ and ​V_source​ are voltages in volts and ​t​ is time in seconds. The time constant ​RC_​ is the product of the circuit's resistance and capacitance, implying that the larger the resistor or the capacitor, the more time it will take to charge or discharge. Its unit is also in seconds.

In the reverse process (when discharging), the equation is similar:

\(V_{capacitor} = V_{0}e^{-t/RC}\)

Where ​_V₀_​ is the initial, charged voltage of the capacitor before it begins discharging.

Because the charge takes time to build up and to release, and that time depends on the properties of the circuit's elements, an RC circuit is useful in many electrical devices that require precise timing. Some common examples are: flash cameras, pacemakers and audio filters.

Example 1: What is the capacitance of a parallel plate capacitor made of two 0.25-m² aluminum plates separated by 0.1 m with Teflon at 20 degrees Celsius?

Given the area of one plate, the separation and the dielectric material, start by looking up the dielectric constant of Teflon. At 20 degrees Celsius, it is 2.1 (remember, it has no units!).

Solving for capacitance:

\(C = \dfrac{κε_0A}{d} \newline C = \dfrac {2.1 (8.854 × 10^{-12}) 0.25}{0.1} \newline C = 4.65 × 10^{-11}\)

Example 2: How long will it take to charge a 100-µF (10^-6 farads) capacitor to 20 V when it is connected to a 30-V battery and in circuit with a 10-kΩ (1,000 Ohms) resistor?

Start by converting the capacitance and resistance to their SI units, and then calculating the RC time constant:

C = 100 µF = 0.0001 F

R = 10 kΩ = 10,000 Ω

RC = 0.0001 F × 10,000 Ω = 1 second

Then, using the formula for a charging capacitor and solving for time ​t​:

\(V_{capacitor} = V_{source}(1-e^{t/RC})
\newline
20 V = 30 V(1-e^{t/1})
\newline
2/3 =1-e^t
\newline
1/3 = e^t
\newline
ln(1/3) = ln(e^t)
\newline
1.1 seconds = t\)

Capacitors and batteries may seem similar as they are both able to store and release electronic charge. But they have several key differences leading them to have different advantages and disadvantages.

First, a capacitor stores energy in a charged electric field while a battery stores energy in chemicals, releasing it via chemical reaction. Because of these material differences, a battery can store more energy than a capacitor of the same size.

However, the chemical reaction needed to release that energy is typically slower than the release of charges through the electric field in a capacitor. So, a capacitor can charge and discharge much more quickly than a battery, providing more electric power in a short spurt. A capacitor is also typically more durable than a battery, making it more environmentally friendly.

For all these reasons, engineers today are seeking to increase the storage limits of capacitors and decrease the charging and discharging times of batteries. Until then, the devices are often used together. For example, a camera's flash and a pacemaker both use a battery and a capacitor to supply long-lasting energy ​and​ deliver it in quick bursts at higher voltages.

Capacitors are often used in circuits to smooth or mediate the voltage changes a device would otherwise experience. For example, most energy delivered to a home comes in an alternating current (AC) supply, which provides a "bumpy" voltage, yet most home appliances require a direct current (DC) supply of energy.

Capacitors in the wall help to transform the signal from AC to DC for these devices. The incoming voltage charges the capacitor, and when it starts to alternate to a lower voltage, the capacitor begins to discharge some of its stored energy. That allows the device on the other side to continue experiencing a more constant voltage than it would without the capacitor.

Capacitors are also useful in devices where certain frequencies of electronic signals might need to be filtered out, say, a radio amplifier or an audio mixer. For example, a capacitor in the circuit can direct low-frequency and high-frequency sounds to different parts of a speaker, such as the sub-woofer or the tweeter. Or, a radio speaker using capacitors to separate frequencies can amplify some but not others, thereby reinforcing the signal of the desired station the radio is tuned into.

​Decoupling in an integrated circuit.​ One of the most ubiquitous uses for a capacitor is in an integrated circuit – the small circuit board containing all the electrical components used to power most consumer electronics, like smartphones. There, the capacitor serves as something of a shield, protecting other electronic components from sudden voltage drops and acting as small, temporary power sources when the supply is momentarily interrupted, as often happens.

Similar to how they help provide direct current to home appliances, capacitors buffer voltage changes for electronics beyond them in the circuit; they "soak up" extra voltage and in turn release their excess voltage when the supply begins to drop.

Decoupling capacitors in integrated circuits specifically remove high-frequency changes to the voltage (since they can absorb some of the voltage change passing through them). This results in the rest of the circuit components experiencing a more even keel of voltage at the levels needed for their correct operation.

​Capacitors as sensors.​ Because capacitor design depends on the materials used, which in turn have differing conductive properties under different conditions, capacitors are important components in electronic sensors.

For example, a humidity sensor uses a dielectric material such as a plastic or polymer that changes its conductance reliably with changing moisture levels. Thus, by reading the conductance across that dielectric, the sensor deduces the relative humidity.

Similarly, some fuel-level sensors, including those in airplanes, use capacitors to gauge how much fuel is left in the tank. In these devices, the fuel itself serves as the dielectric. Once it drops off to a low enough level, the conductivity changes and the pilot is alerted.

Perhaps even more common are capacitive switches used in touchscreen devices. When a person's finger touches a screen, it discharges a small amount of charge, thereby changing the conductance of the device measurably and pinpointed to a specific location. This also explains why wearing gloves interferes with scrolling on a smartphone – the wool or cotton in a glove is a great insulator, keeping the charges in fingers from jumping to the screen.