Capacitors

In October 1745, Ewald Georg von Kleist of Pomerania in Germany found that charge could be stored by connecting a generator by a wire to a volume of water in a hand-held glass jar. Von Kleist's hand and the water acted as conductors and the jar as a dielectric. Von Kleist found that after removing the generator, touching the wire resulted in a painful spark. In a letter describing the experiment, he said "I would not take a second shock for the kingdom of France." The following year, the Dutch physicist Pieter van Musschenbroek invented a similar capacitor, which was named the Leyden jar, after the University of Leyden where he worked. Daniel Gralath was the first to combine several jars in parallel into a "battery" to increase the charge storage capacity.

Benjamin Franklin investigated the Leyden jar, and proved that the charge was stored on the glass, not in the water as others had assumed. Leyden jars began to be made by coating the inside and outside of jars with metal foil, leaving a space at the mouth to prevent arcing between the foils. The earliest unit of capacitance was the 'jar', equivalent to about 1 nanofarad.

Leyden jar or flat glass plate construction was used exclusively up until about 1900, when the invention of wireless (radio) created a demand for standard capacitors, and the steady move to higher frequencies required capacitors with lower inductance. A more compact construction began to be used of a flexible dielectric sheet such as oiled paper sandwiched between sheets of metal foil, rolled or folded into a small package.

Early capacitors were also known as //condensers//, a term that is still occasionally used today. It was coined by Alessandro Volta in 1782 (derived from the Italian //condensatore//), with reference to the device's ability to store a higher density of electric charge than a normal isolated conductor. Most non-English European languages still use a word derived from "condensatore".

A **[|capacitor]** is the [|Fort Knox] of the physicists' and engineers’ world. Capacitors can store and provide an abundant amount of energy that cannot easily be supplied by a traditional battery. They are found in a vast range of different products, but they all serve the same function in each device: to store electrical energy.



The flash bulb on a camera is a prime example of where a capacitor might commonly be used. The battery in the camera is not able to supply enough energy fast enough to cause a quick burst of light to be emitted from the flash bulb. Placing a capacitor in the circuit allows the battery to charge the capacitor with electrical energy. When the situation calls for the flash to be used, the capacitor releases its stored energy, which manifests as a bright flash of light.

media type="youtube" key="5rE-6Uu1CFA" height="344" width="425" Video of a camera capacitor discharging. We can see that capacitors are commonplace in various electronic devices, but how do they work?

All capacitors, regardless of shape or size, are essentially composed of two conducting plates or surfaces that are separated by some distance or radius. These plates become charged when a voltage difference is supplied to the plates. The battery creates an electric field that draws electrons away from the plate connected to the positive battery terminal and in turn causes that plate to become more positively charged. This electric field also pushes electrons from the negative battery terminal to the second plate, causing that plate to become more negatively charged. The surfaces of the capacitor plates are equipotential surfaces, because all points on each plate are at the same potential. This is important because it allows us to ignore the fringe effects that occur on the outer edges of a plate that is not infinitely long. The charging of the capacitor stops once the potential difference between the two plates of the capacitor is equal to the potential difference across the terminals of the battery. Once this equilibrium is reached, the electric field disperses because there are no unbalanced sides and therefore the electrons do not need to be moved through the circuit. The potential difference and the charge on the capacitor are proportional to each other and lead to the following equation: (1) where //C// is the proportionality constant known as the **[|capacitance]** of the capacitor. Its value depends only on the geometry of the plates and not on their charge or potential difference.
 * Capacitance and Charging a Capacitor**

To calculate the capacitance, we must first calculate the electric field using Gauss’ Law: (2) where // ε 0 // is the permittivity constant. This equation can be reduced to Equation 3 if the electric field is uniform and is parallel to the area vector //dA//. (3) Next, we must calculate the potential difference between the capacitor’s plates: (4) Note: Always choose a path for the electric field to follow the electric field lines, from the negative terminal to the positive terminal.
 * Calculating Capacitance**

From the last two equations we can calculate the capacitance of various capacitors of different shapes and designs. A parallel-plate capacitor has two plates of a certain area //A//, separated by a distance //d//. To calculate the potential difference, we enclose the positive plate only with a Gaussian surface and take the following integral: Relating this equation with Equation 1 produces the equation for the capacitance of a parallel-plate capacitor: (5) where //A// is the area of the plate and //d// is the distance between the plates.
 * Parallel-Plate Capacitor**

Using Equation 3 for calculating the charge from the electric field, we can substitute a new area for calculating the capacitance of a cylindrical capacitor: where //r// is the radius of the Gaussian surface and //L// is the length of the Gaussian surface.
 * Cylindrical Capacitor**



Relating this equation to Equation 1 gives an equation for finding the capacitance of a cylindrical capacitor: (6) Where //L// is the length of the capacitor, //a// (//R1// in the diagram) is the radius of the inner cylinder and //b// (//R2//) is the radius of the outer cylinder.



Similarly to the case of a cylindrical capacitor, we can start with Equation 3 and substitute a new area for calculating the capacitance of a spherical capacitor: where //r// is the radius of the Gaussian surface around the capacitor’s center.
 * Spherical Capacitor**



Relating this equation to Equation 1 gives an equation for finding the capacitance of a spherical capacitor: (7) where //a// (//R1// in the diagram) is the radius of the inner sphere and //b// (//R2//) is the radius of the outer sphere.

If a single isolated spherical capacitor exists, we can assume that the single spherical capacitor represents the inner sphere of a capacitor with an infinitely large outer sphere. In this case, the capacitance equation for a spherical capacitor needs to be altered to account for this fact. Assuming the outer sphere of the capacitor is infinite allows us to calculate the capacitance of just the single spherical capacitor we started with. The new equation for the capacitance of a single isolated spherical capacitor is: Replace the radius of the isolated sphere //a// with //R// and let the radius of the outer sphere //b// to approach infinity. This creates a final equation for calculating the capacitance of an isolated sphere capacitor: (8)
 * Isolated Sphere**

Similar to resistors, capacitors can be wired in series and in parallel connections throughout a circuit. Also like resistors, we can replace a group of capacitors with an equivalent capacitor.
 * Capacitors in Parallel and in Series**



When a group of capacitors are wired in series, we can find the value for the equivalent capacitor by finding the inverse of the sum of the inverses of the individual capacitances: (9) A group of capacitors wired in series all carry the same charge across their terminals. Thus, when we relate the potential differences of the capacitors, we can factor the charge out, and are left with the above equation.
 * (a) Calculating Equivalent Series Capacitance**

When a group of capacitors are wired in parallel, we can find the value for the equivalent capacitor by summing the individual capacitances: (10) A group of capacitors wired in parallel all have the same voltage across them. Thus, when we relate the charges of the capacitors, we can factor the voltage out, and are left with the above equation.
 * (b) Calculating Equivalent Parallel Capacitance**

When a capacitor is being charged, an electric field forms between the two capacitor plates and "stores" the potential energy there until the capacitor is discharged. The work required to charge the capacitor is usually provided by the stored energy in a battery connected to the circuit. We can calculate the amount of work done to charge the capacitor and relate the value to the potential energy stored in the electric field by using this equation: Since the work used to charge the capacitor is equivalent to the potential energy stored in the electric field, we can replace //W// with //U//, which leaves us with: (11)We can also express this equation in terms of the potential difference instead of the charge: (12)
 * Calculating Potential Energy Stored in the Electric Field between Capacitors**

If we ignore friction between the capacitor plates, we can find the energy density by assuming the value will be equal at any point in or along the plates. We can calculate the energy density by dividing the potential energy by the volume of the plates, similar to calculating the volume density of any object. If we substitute a different equation for //C//, it leads us to: Lastly, substituting the equation for the electric field, we get the final equation for the energy density: (13) where //E// is the electric field.
 * Energy Density**

Find the capacitance of two concentric metal spheres of radius a and b respectively.
 * Example 1:**

A 100 pF capacitor is charged to a potential difference of 50V, and the charging battery is disconnected. The capacitor is then connected in parallel with a second (initially uncharged) capacitor. If the potential difference across the first capacitor drops to 35V, what is the capacitance of the second capacitor?
 * Example 2:**

Given a 7.4 pF air-filled capacitor, you are asked to convert it to a capacitor that can store up to 7.4 µJ with a maximum potential difference of 652V. Which dielectric should you use to fill the gap in the capacitor if you do not allow for a margin of error? This reveals that the best dielectric to use is Pyrex.
 * Example 3:**

And now, here's something fun to do with a capacitor:

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Plasma lamps are high frequency electricity capacitors:

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