no tagsLecture 012: Electrostatic Energy and CapacitorsSteveSekula, 21 September 2010 (created 15 September 2010)Goals of this Lecture:Introduce the idea of energy stored in the electric field (electrostaticenergy)Introduce the idea of a "capacitor", capacitance, and discuss simplecapacitorsElectrostatic energyThe electric field can not only do work, but store energy. The storage ofenergy in electric fields, and its release, is the basis of chemical energy.The metabolizing food and burning fuel are basically the act of rearrangingconfigurations of electric charge so as to release some of the energy storedin the original arrangement. The example we'll start with today seemssimple, but it's deceptively powerful because it contains the essentialelements of any system that stores energy in the electric field.This exercise will not only introduce the idea of energy stored in anelectric field, but will help to review some of the basic concepts we startedlearning a few weeks ago.Imagine that I place a point charge, , in a region of empty space freefrom any electric fields or other forces. How much work does it take toplace at a point in such a region free of electric fields and other forces?ANSWER: none. Absent any source of external force, we can placecharge at a point at no cost in energy. Remember that -if , then no work is needed.General Physics - E&M (PHY 1308) LectureNotesGeneral Physics - E&M (PHY 1308) LectureNotesq 1> 0q 1q 1W r AB=RF~Á d~F ~= 0General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...1 of 4 09/21/2010 10:05 PMNow, let's imagine that I want to bring a second positive charge, , fromvery far away up to a distance from . I want to figure out the workrequired to do that. Let's attack this from the perspective of electricpotential difference:If we are going to bring another charge from infiniity to a distance from , then:So the work required to bring that charge to a point from is:Now we bring in a third positive charge, , also from very far away(infinity). We have to do work to get it there, against the repulsion of theother two charges. If we also place the third charge a distance from eachof the other two, forming an equilateral triangle, then we need to do work:So the total work required to form this configuration of charges is:Because electric forces are conservative forces, the work done to make theconfiguration is equal to the energy stored in the electric field. It takesenergy to hold the charges together, to keep them from flying apart; thatmeans energy is stored in the configuration.Releasing any of the charges converts stored energy in the field intokinetic energy in one or more of the electric charges. Energy stored in anelectric field can be released in other forms of energy, such as kineticq 2jrj = a q 1ÁV AB= VBÀ VAq 2a q 1ÁV kq =r r;1= À1a q 1W q V (a) q q =a ABÑ W2= À2= k1 2q 3a W q q =a q q =a 3= k1 3+ k2 3W q q =a q q =a q q =a = W1+ W2+ W3= 0 + k1 2+ k1 3+ k2 3General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...2 of 4 09/21/2010 10:05 PMenergy.It doesn't matter in what order we assemble this simple example; theenergy stored is the same. If one of them had been negative, it would takework to separate that charge from the other two due to its attraction.Electrostatic energy can be positive or negative, depending on whether ornot it took work to assemble the charges in the first place.This example is a simple metaphor for a molecule, such as a watermolecule. In fact, in water the electrostatic energy is negative and it takesan injection of work (costs energy) to separate the hydrogen from theoxygen. For water, electrostatic energy is NEGATIVE. Equivalently, that isthe energy released when water forms from individual atoms.CapacitorsSo we can store energy in an electric field. A device that does this is calleda capacitor. Specifically,A capacitor is a pair of electrical conductors that carry equal butopposite chargesThe two conductors are thus attracted to one another, and it takes work tokeep them apart. The easiest to analyze capactor is the parallel-platecapacitor, although capacitors come in many configurations.Understanding the parallel-plate capacitor will give us insight intoelectrostatic energy and the electric field.Parallel plate capacitorA device with two parallel thin sheets of conductor. Charge is removedfrom one plate and added to the other (e.g. a battery can do this). Thus wehave equal but opposite charges on the two plates, and close to the centerof the plates we can understand the electric field by modeling the systemwith two infinite thin sheets of opposite charge. The field lines areperpendicular to the sheets and go from positive to negative.Closer to the ends, the field becomes nonuniform. But we can neglect thisand still get tremendously far in understanding these devices.General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...3 of 4 09/21/2010 10:05 PMIf you do the Gaussian Surface trick and apply Gauss's Law to either of theplates, you'll find that the electric field at the surface of the conductor isgiven by . If we've spread out charge uniformly over the twoplates, then for either plate , where is the area of the plate. Thusthe uniform field between the two is:Do a demonstration of this using two parallel plates and a GaussianSurface. Analyze the electric field between two parallel plates and showthat it is and NOT . Remember, these are conductors andthe opposite charges accumulate on the faces closest between the twoplates.The potential difference between the two plates issince the field is uniform.jEj =Ï = Û0Û =A = Q A E =Ï A = Q0E =Ï = Û0E Û=Ï = 20V d d=Ï A = E = Q0General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...4 of 4 09/21/2010 10:05
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