Capacitor

This page is about Capacitors and how they work.

The Main Idea

A capacitor is made up of two uniformly charged disks. It is able to store electricity in an electric field. They are able to continue the functions of electronics for a short time while they are unplugged. They essentially are able to act like a power supply by storing electricity.

Mathematical Model

[math]\displaystyle{ \ C=\frac{q}{V} }[/math]

[math]\displaystyle{ \ E=\frac{Q/A}{\epsilon_0 } }[/math] One plate has charge [math]\displaystyle{ \ +Q }[/math] and other plate has charge [math]\displaystyle{ \ -Q }[/math]

Theory of operation

Overview

A capacitor is made of two conductors separated by a non-conductive area. This area can be a vacuum or a dielectric (insulator). A capacitor has no net electric charge. Each conductor holds equal and opposite charges. The inner area of the capacitor is where the electric field is created.

Hydraulic analogy

Charge flowing through a wire is compared to water through a pipe. A capacitor is similar to a membrane blocking the pipe. The membrane can stretch but does not allow water (charges through). We can use this analogy to understand important aspects of capacitors: Charging up a capacitor stores potential energy, the same way a stretched membrane has elastic potential energy. As the capacity of a capacitor decreases the voltage drop increases. It resists the current flow as it is charged up. The more water stretching the membrane, the harder it is to stretch further. The higher the capacitance leads to a higher capacity. A stretchier membrane allows for more water before it is done stretching. The current affects the charge on a capacitor. As one side of the capacitor is charged up, the other side loses charge. When a certain amount of water pushes to another side of the membrane the side they came from lost as much water as the new side gained.

Capacitance

Capacitance is the ability of something to store a charge. This is important to a capacitor and allows us to measure how effective it is. The higher the capacitance number is the more charge a capacitor can hold.

Capacitance in a circuit is found by the following:

[math]\displaystyle{ \ C=\frac{q}{V} }[/math]

Electric Field

Two charged plates separated by very small gap s

Electric Field of two uniformly charged disks: A Capacitor

Electric field near the center of a two-plate capacitor

[math]\displaystyle{ \ E=\frac{Q/A}{\epsilon_0 } }[/math] One plate has charge [math]\displaystyle{ \ +Q }[/math] and other plate has charge [math]\displaystyle{ \ -Q }[/math]; each plate has area A; Direction is perpendicular to the plates. Assumption: separation between capacitor is very small compared to the area of a plate.

Fringe Field (just outside the plates near center of disk)

[math]\displaystyle{ \ E_{fringe}=\frac{Q/A}{2\epsilon_0 }(\frac{s}{R}) }[/math] [math]\displaystyle{ \ s }[/math] is the separation between plates; [math]\displaystyle{ \ R }[/math] is the radius of plate

The Algorithm

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Two charged plates separated by very small gap s

Step 1. Cut up the charge distribution into pieces and find the direction of [math]\displaystyle{ \Delta \vec{E} }[/math] at each location

Approximate electric field of a uniformly charged disk [math]\displaystyle{ \ E=\frac{Q/A}{2\epsilon_0 }[1-\frac{s}{R}] }[/math] or [math]\displaystyle{ \ E=\frac{Q/A}{2\epsilon_0 } }[/math]

At location 2, midpoint between two disks, both disks contribute electric field in the same direction. Therefore, [math]\displaystyle{ \vec{E}_{net} }[/math] is the largest at this location.

At location 1, because negative charged plate is closer, [math]\displaystyle{ \vec{E}_{net} }[/math] is to the right.

At location 3, because positive charged plate is closer, [math]\displaystyle{ \vec{E}_{net} }[/math] is to the left.

Step 2. Find the electric field of each plate

Assumption: Ignore the electric field due to the small charges on the outer surface of the capacitor since it's very small; Assume that separation between capacitor is very small compared to radius or a disk; consider that location 1 and 3 are just near the disk

To make equation valid at all locations, choose origin at the inner face of the left disk so Electric field of the negative charged plate is [math]\displaystyle{ \ E_{-}=\frac{Q/A}{2\epsilon_0 }[1-\frac{s}{R}] }[/math] to the left and Electric field of the positive charged plate is [math]\displaystyle{ \ E_{+}=\frac{Q/A}{2\epsilon_0 }[1-\frac{s-z}{R}] }[/math] to the left

Step 3. Add up

Location 2: Therefore, the location 2, middle of the capacitor, is located z from the negative charged plate and s-z from the positive plate. Since they are in same direction, we simply add them to find [math]\displaystyle{ \vec{E}_{net} }[/math]

[math]\displaystyle{ \vec{E}_{net} = \ E_{-} + \ E_{+} =\frac{Q/A}{2\epsilon_0 }[1-\frac{z}{R}] + \frac{Q/A}{2\epsilon_0 }[1-\frac{s-z}{R}] = \frac{Q/A}{\epsilon_0 }[1-\frac{s/2}{R}] = \frac{Q/A}{\epsilon_0 } }[/math]

Location 1: At location 1, we can see that [math]\displaystyle{ \vec{E}_{net} = \ E_{-} - \ E_{+} = \frac{Q/A}{2\epsilon_0 }[1-\frac{z}{R}] - \frac{Q/A}{2\epsilon_0 }[1-\frac{z+s}{R}] = \frac{Q/A}{2\epsilon_0 }(s/R) }[/math]

Location 3: At location 1, we can see that [math]\displaystyle{ \vec{E}_{net} = \ E_{+} - \ E_{-} = \frac{Q/A}{2\epsilon_0 }[1-\frac{z-s}{R}] - \frac{Q/A}{2\epsilon_0 }[1-\frac{z}{R}] = \frac{Q/A}{2\epsilon_0 }(s/R) }[/math]

Step 4. Check Unit

[math]\displaystyle{ \frac{C/m^2}{N*m^2/C^2} = \frac{N}{C} }[/math]

[math]\displaystyle{ (\frac{C/m^2}{N*m^2/C^2})(m/m) = \frac{N}{C} }[/math]

Examples

Middling

For a two-disk capacitor with radius 50cm with gap of 1mm what is the maximum charge that can be placed on the disks without a spark forming? Under these conditions, what is the strength of the fringe field just outside the center of the capacitor? The air breaks down and a spark forms when the electric field in air exceeds 3e6 N/C

First, convert lengths to meter

[math]\displaystyle{ 50cm = 0.5 m }[/math]

[math]\displaystyle{ 1mm = 0.001mm }[/math]

Second, set up the equation [math]\displaystyle{ 3e6 N/C= \frac{Q/A}{\epsilon_0 } }[/math]

Third, solve for Q

[math]\displaystyle{ A= 0.5*0.5*3.14 = 0.785 }[/math]

[math]\displaystyle{ Q= 3e6*8.85e-12*0.785 = 2.01e-5 }[/math]

[math]\displaystyle{ E_(fringe) = \frac{Q/A}{2\epsilon_0 }(s/R) }[/math]

[math]\displaystyle{ \frac{(2.01e-5)/(0.785)}{2\epsilon_0 }(0.001/0.5) = 3000 N/C }[/math]

Simple

You have a circuit consisting of a 10V battery and three capacitors with capacitances of 7muF, 4muF, and 6muF. Find the equivalent capacitance of the three capacitors connected in series with the battery.

[math]\displaystyle{ \frac{1}{C}=\frac{1}{7uF}+\frac{1}{4uF}+\frac{1}{6uF} }[/math]

[math]\displaystyle{ \frac{1}{C}=0.143+0.25+0.167 }[/math]

[math]\displaystyle{ \frac{1}{C}=0.56 }[/math]

[math]\displaystyle{ C =\frac{1}{0.56} }[/math]

[math]\displaystyle{ C=1.79uF }[/math]

Connectedness

1. How is this topic connected to something that you are interested in?

Batteries use capacitor to control circuit. Capacitors are used in almost all electronics. If you have any interest in electronics then capacitors apply to you.

2. How is it connected to your major?

Capacitor and electronic means of Power Factor Correction provide well-known benefits to electric power systems. These benefits include power factor correction, poor power factor penalty utility bill reductions, voltage support, release of system capacity, and reduced system losses. A high power factor signals maximum use of electrical power, while a low power factor leads to purchasing more power to obtain the same load kW, which you pay for in various ways on your utility bill.

3. Is there an interesting industrial application?

A capacitor can store electric energy when it is connected to its charging circuit. And when it is disconnected from its charging circuit, it can dissipate that stored energy, so it can be used like a temporary battery. Capacitors are commonly used in electronic devices to maintain power supply while batteries are being changed.

History

The first capacitor was created in 1745 by a man named Ewald Georg von Kleist. He was from Pomerania, Germany. He connected a generator to a wire and ran it to a glass jar lined with metal foil and filled with water. This was a rough idea of a capacitor and was not able to store electricity yet. Rather, it proved a concept. In 1746, Pieter van Musschenbroek, a Dutchman, created a similar capacitor and named if the Leyden jar. Both men noticed a much larger shock from the generator when it was connected to a jar of waster. The current issue was that the storage capacity of the jar was not big enough and it discharged almost immediately. Later on, Daniel Gralath connected multiple jars together in parallel to increase the storage capacity. Benjamin Franklin found that the charge was being stored in the glass and not the water. He also coined the term "battery" as it is used in this capacity. The Leyden jar was used until 1900 when smaller capacitors with larger capacities were needed. Capacitors saw many forms before reaching what they are today. Today they are made with a dielectric (insulating materials) sandwiched between metal plates.

See also

Internet resource for capacitor: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/pplate.html

References

Matter and Interactions, 4th Edition http://www.eaton.com/Eaton/ProductsServices/Electrical/ProductsandServices/PowerQualityandMonitoring/PowerFactorCorrection/index.htm http://hyperphysics.phy-astr.gsu.edu/hbase/electric/pplate.html