(3.5.1) V E x, E, (3.5.2)


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1 Lecture 3.5 Capacitors Today we shall continue our discussion of electrostatics and, in particular, the concept of electrostatic potential energy and electric potential. The main example which we have been considering so far was the example of the charged pointlike object inside of the uniform electric field of the parallelplate capacitor. In this situation there is electrostatic force acting on the object from the field. So, if one releases this charge, it will start moving and the electric field of the capacitor will perform work on the charge transferring potential energy of electric field into kinetic energy of the moving charge. So, the capacitor is capable of storing potential energy U, which can later be used for accelerating of a charge. This potential energy depends on configuration (or arrangement) of the system which means that it depends on the design of the capacitor. We have also introduced the concept of electric potential. The change of electric potential is defined as the change of electric potential energy per unit positive charge. It is similar to the concept of electric field, where the field itself does not depend on the testing charge, the electric potential depends on configuration of the system but it does not depend on the absolute value of the testing charge. For instance, we have proved the relationship between the electric field (which only depends on the charge density of capacitor s plates) of the parallelplate capacitor and the potential difference between its plates. If we have a testing charge q moving in the direction of the electric field along the xaxis perpendicular to the pates of the capacitor then W qex V Ex, q q (3.5.1) E x As we said, this means that electric field is the rate of change of electric potential. We know that the constant value of the electric field inside of the plane capacitor is E, (3.5.2)
2 If the distance between the plates of the parallelplate capacitor is d, which is a fixed value for any given capacitor then we can calculate the potential difference between the plates combining equations and as d E Ed, d Qd A (3.5.3) So, we can see that potential difference between the plates of the parallelplate capacitor is only dependent on its characteristics, such as the distance between the plates, the area of each plate and charge placed on the pates. For a given capacitor the values of d and A are fixed. So, if you provide a certain potential difference across the plates of the capacitor (for instance by connecting it to the battery) then the capacitor will be charged. It will hold the charge of absolute value Q (positive on one plate and negative on the other plate), which is Q A d (3.5.4) Or if the given value of electric charge is placed on the plates of the parallelplate capacitor, then there will be a certain value of potential difference across the plates defined by the equation A capacitor is the electric device which can store electric energy or electric charge. So, the better the capacitor is the more charge you can store in it. But for any given capacitor and given potential difference this amount of charge is limited by the equation Capacitor gets its name because of the capacity to store energy and charge. One can introduce a special physical quantity, which shows this capacity for a given potential difference, which is Q C, (3.4.5) The quantity C is called capacitance of the capacitor and it is measured in special units called Farad (F), 1F=1C/1V. Even though the example, which we just have considered, was the example of the parallelplate capacitor, but the fact that the charge is proportional
3 to the potential difference between the plates (equation 3.5.4) is general. Therefore, one can introduce capacitance for any type of the capacitor by means of its definition In the case of the parallelplate capacitor we just showed that A C, (3.4.6) d It is obvious that the larger area of the plates allows more space for the charge, so C gets bigger. Increase of the distance between the plates causes the decrease of the potential change rate and as a result the decrease of the electric field, so the smaller charge is needed to provide this smaller field. More complicated shapes of the capacitors, such as cylindrical capacitor or spherical capacitor, are often used in practice. In the case of those capacitors C depends on A and d in more complicated way than the one described by equation 3.4.6, but the general conclusion that it depends on area of the plates and separation distance between the plates is still true. In order to find capacitance for these more complicated systems, one shall start with the Gauss law to figure out the electric field inside of the capacitor. This task is similar to several examples of using Gauss law for cylindrical and spherical symmetry, which did in the past. After the electric field is found, one shall use the relation between the electric field and electric potential V f E ds, (3.5.7) i to determine the potential difference between the capacitor s plates and then definition to calculate capacitance. We shall consider several examples for different configurations during the lecture. For practical purposes to store as much charge as possible not just one but several capacitors can be used. If several capacitors are used, there are alternative ways of how you can connect them to the battery. Once again we shall consider a parallelplate capacitor consisting of two identical metal plates with area A separated distance d. We connect this capacitor in a circuit with a battery and a switch. When the switch is open, there is no excess charge on either plate. The switch is then closed. If this capacitor is connected to the battery, the potential difference across the capacitor, V, will become the same as the one provided by the battery. This usually happens not immediately but after some relatively short time as the
4 capacitor comes to equilibrium. At the end there is no electric field outside of the capacitor but only inside. And because of that the total charge on the capacitor will be zero, which means both plates have equal in magnitude but opposite in signs charges. The plate connected to the negative terminal of the battery will have the negative charge. The palate connected to the positive terminal will have the positive charge. This is because the positive terminal attracts free electrons from one plate and the negative terminal repels the electrons from the other plate. We shall start from the case where several capacitors with different values of C are all connected to the same battery in parallel. The good way to think about two capacitors connected in parallel is to notice that the voltage provided by the battery is the same across both capacitors, while the total charge on the plates is just the sum of the charges on each capacitor. This means that the effective capacitance will be C Q Q Q V V Q Q C1 C2 V V. (3.5.8) At the same time when considering capacitors in series, one can notice that the total voltage across these capacitors can be added up from the voltages on each of them, while the effective charge is the same for each of the capacitors, since all intermediate plates are neutral in total. This means that Q Q 1 1 C V V V V Q V Q 1 C 1 C C C C , (3.5.9) As we just have seen the capacitance depends on the shape and the size of the capacitor, however, we have only limited our attention by the case when the space between the plates is empty. In reality it will always be filled with some substance. The purpose of the capacitor is to hold as much charge as possible. To achieve this goal we can insert dielectric material between the plates of the capacitor. Every time when this dielectric material is inside of the external electric field of the capacitor it will be polarized. This means that electric dipoles (molecules of dielectric) will be oriented in such a way that their negative sides will be directed towards the positive plate of the capacitor and their positives sides will be directed towards negative plate of the capacitor. Thus the total electric field of all the diploes is oriented in such a way that it is directed
5 opposite to the external field of the capacitor which has caused that orientation. So, the total field inside becomes less compared to what it was before the dielectric was inserted. One can introduce a special quantity known as dielectric constant (or electric permittivity of dielectric), which shows by how much electric field in dielectric is smaller compared to the electric field in vacuum, so that E, (3.5.1) E where E stands for the electric field in dielectric and E for the electric field in vacuum. Value of dielectric constant depends on material. Those values are listed in table 25.1 on page 67 in the book. This phenomenon can be explained, by appearance of additional induced charge inside of the dielectric which has the opposite sign to the original charge, so we can also say that the effective charge enclosed between the plates of the capacitor is now qeff q and the Gauss law should now read q q, eff E da which we can also rewrite as DdA q, (3.5.11) where D E is known as the vector of electric displacement, but the charge is still the free charge only, and the induced charge is ignored in the righthand side of this equation. As a result of having dielectric inside of the capacitor the effective electric field reduces causing the effective reduce of the potential difference by the same factor of, so foe the parallelplate capacitor we have Q A C / d (3.5.12) This means that capacitance gets higher if some dielectric is inserted inside of the capacitor. However, this does not set the limit on how large the charge stored can be. If you continue to increase potential difference between the plates of the capacitor, the molecules of dielectric can be broken apart by strong enough external electric field. Then
6 the dielectric becomes a conductor and electric charge flows from one plate to another plate of the capacitor. So, we say that dielectric break down has occurred. Let us now find the energy stored inside of the capacitor. This energy storage in the capacitor takes place in the process of its charging, when the charge is transferred from one of the capacitor s plates to another plate until the potential difference between the plates reaches a given value (for instance one provided by the battery). During this process both charges on the plates and potential difference between the plates are changing. The charge built up on the plates is related to the current value of the potential difference. Since C is constant for a given capacitor then charge increases as a linear function of increasing potential difference. For every small charge dq transferred from one plate to another plate the potential energy of the system increases by a small amount du Vdq. To find the total energy stored in the capacitor, one has to perform a summation of all these small energies which means to find the area under q, graph or to integrate potential as function of charge, which is Q U QV CV C 2 2. (3.5.13)
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