ALPHY6: CAPACITORS

HISTORY OF CAPACITORS

Nearly everyone is familiar with the static charge generated by friction — a phenomenon formally known as triboelectricity. Walking across a carpeted floor, combing one’s hair on a dry day, or pulling the transparent tape off a roll all result in the separation of small amounts of positive and negative charge.

The earliest known written account of charging by friction goes back as far as the Sixth Century BCE when the Greek scientist Thales of Miletus (635–543 BCE) noted that amber rubbed with animal fur acquired the ability to pick up small bits of material.

For roughly the next 2300 years, wherever electricity was studied, somebody had to take two different materials and rub them together to create separated islands of positive and negative charge.

Fast forward to Eighteenth-Century Europe, an era known as the Enlightenment, a time and place characterized by the expansion of culture and the acquisition of knowledge. Among the empowered and educated classes of the Enlightenment, science was a fashionable pursuit and lectures on scientific subjects were well attended. Those given by electricians were among the most popular. (The word electrician originally referred to a person knowledgeable in the nature of static electricity.) Electricity was a hot topic in the Eighteenth Century and much exploration was being done with electrostatic machines that generated a charge by friction.

While friction is an easy and inexpensive means to separate charge for use in electric experiments, the amounts of charge available are quite small. If electricity was going to be anything other than an irritating side effect of walking across a carpet, some means for increasing the amount of charge available for experiments had to be found.

The first device for storing charge was discovered in the winter of 1745–46 by two electricians working independently: Ewald von Kleist (1715–1759), dean of the cathedral at Kammin, Prussia (now Kamień, Poland), and Pieter van Musschenbroek (1692–1761), professor of mathematics and physics at the University of Leyden in Holland (now spelled Leiden).

The device built by von Kleist consisted of a medicine bottle partly filled with water and sealed with a cork. A nail was pushed through the cork and into the water. Holding the bottle in one hand, the nail was then brought in contact with the terminal of an electrostatic machine allowed to acquire some charge.

When von Kleist reached for the nail to remove it from the stopper while still holding the bottle the separated charges were able to reunite by flowing through his body. Van Musschenbroek’s device and experiences with it were almost the same as von Kleist’s, but with three major exceptions. First, a visiting student Andreas Cunæus (1712–1788) made the shocking discovery not van Musschenbroek himself; second, he made many improvements to the device (most importantly, removing the water and wrapping the inside and outside of the jar with metallic foil); and third, he wrote his colleagues to tell them all about it.

A capacitor is a device used to store electrical charge and electrical energy. It consists of at least two electrical conductors separated by a distance. (Note that such electrical conductors are sometimes referred to as “electrodes,” but more correctly, they are “capacitor plates.”) The space between capacitors may simply be a vacuum, and, in that case, a capacitor is then known as a “vacuum capacitor.” However, the space is usually filled with an insulating material known as a dielectric. (You will learn more about dielectrics in the sections on dielectrics later in this chapter.) The amount of storage in a capacitor is determined by a property called capacitance, which you will learn more about a bit later in this section.

Capacitors have applications ranging from filtering static from radio reception to energy storage in heart defibrillators. Typically, commercial capacitors have two conducting parts close to one another but not touching, such as those in Figure \(\PageIndex{1}\). Most of the time, a dielectric is used between the two plates. When battery terminals are connected to an initially uncharged capacitor, the battery potential moves a small amount of charge of magnitude \(Q\) from the positive plate to the negative plate. The capacitor remains neutral overall, but with charges \(+Q\) and \(-Q\) residing on opposite plates.

A system composed of two identical parallel-conducting plates separated by a distance is called a parallel-plate capacitor. The magnitude of the electrical field in the space between the parallel plates is \(E = \sigma/\epsilon_0\), where \(\sigma\) denotes the surface charge density on one plate (recall that \(\sigma\) is the charge Q per the surface area A). Thus, the magnitude of the field is directly proportional to Q.

Capacitors with different physical characteristics (such as shape and size of their plates) store different amounts of charge for the same applied voltage \(V\) across their plates. The capacitance \(C\) of a capacitor is defined as the ratio of the maximum charge \(Q\) that can be stored in a capacitor to the applied voltage \(V\) across its plates. In other words, capacitance is the largest amount of charge per volt that can be stored on the device:

\[C = \frac{Q}{V} \label{eq1}\]

The SI unit of capacitance is the farad (\(F\)), named after Michael Faraday (1791–1867). Since capacitance is the charge per unit voltage, one farad is one coulomb per one volt, or

\[1 \, F = \frac{1 \, C}{1 \, V}.\]

By definition, a 1.0-F capacitor is able to store 1.0 C of charge (a very large amount of charge) when the potential difference between its plates is only 1.0 V. One farad is, therefore, a very large capacitance. Typical capacitance values range from picofarads (\(1 \, pF = 10{-12} F\)) to millifarads \((1 \, mF = 10^{-3} F)\), which also includes microfarads \((1 \, \mu C = 10^{-6}F)\).. Capacitors can be produced in various shapes and sizes

ALPHY6: CAPACITORS ASSIGNMENT ALPHY6: CAPACITORS ASSIGNMENT

• A capacitor is a system of two insulated conductors.

• The parallel plate capacitor is the simplest example.  When the two conductors have equal but opposite charge, the E field between the plates can be found by a simple application of Gauss’s Law.

Assuming the plates are large enough so that the E field between them is uniform and directed perpendicular, then applying Gauss’s Law over surface S₁ we find,

where A is the area of S₁ perpendicular to the E field and σ is the surface charge density on the plate (assumed uniform).  Therefore,

everywhere between the plates.

• The potential difference between the plates can be found from

where A and B are points, one on each plate, and we integrate along an E field line, d is the plate separation, A the plate area and q the total charge on either plate.

• The capacitance (capacity) of this capacitor is defined as,

• The expression for C for all capacitors is the ratio of the magnitude of the total charge (on either plate) to the magnitude of the potential difference between the plates.

• Units of C:       Coulomb/Volt = Farad,    1 C/V = 1 F

 Note that since the Coulomb is a very large unit of charge the Farad is also a very large unit of capacitance.  Typical capacitors in circuits are measured in μF (10^-6) or pF (10^-12).

•  Note that the expression for the capacitance of the parallel plate capacitor depends on the geometric properties (A and d).  Even though it appears that there is also a dependence on the charge and potential difference (q/ΔV), what happens is that whatever charge you place on the capacitor the pd adjusts itself so that the ratio  q/ΔV remains constant.   This is a general rule for all capacitors.  The capacitance is set by the construction of the capacitor – not the charge or voltage applied.

•  The above expression for the parallel plate capacitor is strictly only true for an infinite parallel plate capacitor – in which “fringing” (see above) does not occur.  However, so long as d is small compared to the “size” of the plates, the simple expression above is a good approximation.

•  The parallel plate capacitor provides an easy way to “measure” ε₀

• As indicated above the parallel plate capacitor is the most basic capacitor.  You should also be able to determine the expressions for the capacitance of spherical and cylindrical capacitors,

                                                        

Energy and Capacitors

• One of the most important uses of capacitors is to store electrical energy.

If a capacitor is placed in a circuit with a battery, the potential difference (voltage) of the battery will force electric charge to appear on the plates of the capacitor.  The work done by the battery in charging the capacitor is stored as electrical (potential) energy in the capacitor.  This energy can be released at a later time to perform work.

• The work necessary to move a charge dq onto one of the plates is given by, dW = Vdq, where V is the pd (voltage) of the battery (= q/C).  The total work to place Q on the plate is given by,

which is equal to the stored electrical potential energy, U.

• The electrical energy actually resides in the electric field between the plates of the capacitor.  For a parallel plate capacitor using  C = Aε₀/d and  E = Q/Aε₀ we may write the electrical potential energy,

(Ad) is the volume between the plates, therefore we define the energy density,

• Although we have evaluated this expression for the energy density for a parallel plate capacitor it is actually a general expression.  Wherever there is an electric field the energy density is given by the above.

Combinations of Capacitors

It is common to find multiple combinations of capacitors in electrical circuits.  In the simplest situations capacitors can be considered to be connected in series or in parallel.

• • Capacitors in Series

When different capacitors are connected in series the charge on each capacitor is the same but the voltage (pd) across each capacitor is different

In this situation, using the fact that V = V₁ + V₂ +V₃  we can show that, as far as the voltage source is concerned, the capacitors can be replaced by a single “equivalent” capacitor C_eq  given by,

• • Capacitors in Parallel

For capacitors connected in parallel it is the voltage which is same for each capacitor, the charge being different.

Using the fact that Q_Total= Q₁ + Q₂ + Q₃ we can show that the equivalent capacitor, C_eq  is given by,

</blockquote
HOW CAPACITORS WORK