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Module II Lesson 4 Capacitors Capacitors are important components of electronic circuits and of electrical machinery and power grids. You can find large oil-insulated capacitors on power line poles or small ceramic-insulated capacitors in a radio. In each application, the capacitor is used to store electrical charge and electrical energy; energy may be stored for a short time in an alternating-current cycle, or for a longer time until the energy is needed, as in a strobe light for a camera. Your body can be a capacitor, storing up enough charge and energy to cause a painful spark when you “discharge” yourself to another conductor. Practical capacitors are basically two conducting plates or sheets separated by an insulator (dielectric) such as air, oil, paper, plastic film, or even the oxide layer on one of the conducting surfaces. This lesson will treat the basic physics of capacitors. 4-1: Definition and Calculation of Capacitance OBJECTIVES: Define the terms “capacitance” and use these definitions to relate capacitance, voltage difference, and charge in a capacitor. Derive and use expressions for the capacitance of capacitors that have planar, cylindrical, or spherical symmetry. PREREQUISITE: Determining electric fields and potentials for charge distributions with planar, cylindrical, and spherical symmetries Reading Assignment Study in your textbook Chapter 24, Section 1. Commentary Once again we have an important physical quantity defined as a constant of proportionality in a simple equation. In the equation (1) notice that the capacitance, , is a quantity that is characteristic of a particular capacitor- which we can think of as a pair of insulated conductors. We can evaluate for any capacitor by placing equal but opposite charges of magnitude conductors and measuring the voltage difference between them. on the two The charge is sometimes called the “charge on the capacitor” even though the capacitor as a whole, of course, has a net charge of zero. Obviously, if we know any two of the quantities in Eq. (1), it is possible to solve quickly for the third. EXAMPLE 4-1: An initially uncharged difference of capacitor is charged to a potential by transferring charge from one plate of the capacitor to the other. How much charge was transferred? Solution: Capacitance is defined as the ratio or the magnitude of the charge on one or the other of the conductors to the potential difference between them: (the larger the capacitance, the more charge the capacitor will hold for a given potential difference). The answer is therefore . Capacitance, then does not depend on the charge applied to a capacitor or the resultant potential difference; rather, it determines the relation between these quantities. For now, we are concerned with capacitors whose plates are separated by a vacuum (or air), and in this case the capacitance depends only on the geometry of the “plates” that form the capacitor. Gauss’s law provides a means for us to calculate the capacitance of certain geometric plate configurations. To do this we first imagine that the plates carry charges of magnitude , and then use Gauss’s law (or some other method) to find the electric field between the plates. Next, we must integrate to find the magnitude of the potential difference between the plates. (We are concerned only with because capacitance is always positive.) Finally, we use the definition to find . This procedure is illustrated in Section 24.1 of the text and the following additional examples. (We recommend that you not try to memorize formulas for the capacitance of particular capacitors; instead, you should concentrate on being able to derive these formulas.) EXAMPLE 4-2: A spherical capacitor consists of two concentric spherical shells of radii and ( ). Determine its capacitance. Solution: To derive the capacitance, we imagine that the spheres carry opposite charges of magnitude . We know that the potential due to the outer sphere is constant in the space between the conductors so that the total potential in the region depends on the distance from the center of the capacitor as . Therefore the potential difference between the conductors is . From the definition of capacitance, , we obtain immediately . This problem is of a type you can solve without direct application of Gauss’s law- to the extent that you can remember the forms of the potentials due to particular charge distributions (e.g., Figure 23.17 in the text). EXAMPLE 4-3: The cylindrical capacitor at right is long and consists of two conducting cylinders that are coaxial as shown. Determine the capacitance of this capacitor and compute the potential difference if the inner conductor carries a charge of . ( .) Solution: Before reading this solution, make sure for yourself that you can derive the expression for the capacitance. In the space between the conductors, there is no field due to the outer cylinder so that for a charge on the inner conductor, you can determine from Gauss’s law (do it!) that the electric field in this region is given by where is the length of the cylinder. We integrate this expression to find the potential difference and obtain . Therefore, using the given data, / and . If the gap between the capacitor plates is small compared to their other dimensions, as in the preceding example, then it is reasonable to ignore edge effect (“fringing”); that is, to treat the plates as infinite rather than correcting for the field curvature that occurs at the boundaries of a finite capacitor. As you reread the text assignment and study Sample Problems 1 through 4, take some time to consider the relation of the unit of capacitance, the farad, to other SI units. Sample Problem 1 should give you an idea of the magnitude of the farad, and help to explain why and are more commonly seen in units. (The example may be somewhat misleading, however. Since capacitance is inversely proportional to plate gap, it is possible to package of capacitance into a much smaller space by making the gap between the plates very small and, as we shall see later, filling it with a dielectric.) Practice Exercise 4-1 Write your solutions to the following problems in your notebook. 1. A capacitor (initially uncharged) is charged through the transfer of electrons from one plate to the other. What is the resulting potential difference between the plates? 2. A capacitor is made of two flat, circular, conducting plates of radius separated in a vacuum by , . Determine the capacitance. Check your answers with the Module 2 Answer Key and review Section 4-1 as necessary before going on. 4-2: Connected Capacitors OBJECTIVE: Determine the equivalent capacitance of a set of capacitors connected together and determine the charge and voltage on each capacitor of the set. Reading Assignment Study in your textbook Chapter 24, Section 2. Commentary This section provides a point of entry into the sizable topic of electrical devices as elements in circuits. For one thing, it introduces a schematic notation that allows us to easily represent a particular capacitor (or set of capacitors) charged to a given potential difference. We also encounter the notion that a set of connected capacitors is equivalent, as a circuit element, to a single capacitor of appropriately chosen capacitance. (This idea will be extended to other circuit elements in later lessons.) There are two basic ways that capacitors (or other circuit elements) can be connected together- in series and in parallel- and these give rise to two separate cases in the calculation of equivalent capacitance. The case of capacitors in series or parallel is treated in the text. Example 4-4 below also shows how to derive the equivalent capacitance for a parallel connection. EXAMPLE 4-4: Diagram (a) below shows three capacitors- and - connected in parallel (i.e., the plates of each capacitor are connected by wires to the same terminals, and ). Determine the equivalent capacitance for this arrangement. In other words, determine the single capacitance [ in diagram ] that will give the same potential difference when a given charge transferred from terminal and distributes itself onto the to terminal is plates. Solution: Let charges on charge , , and be , , and , respectively. The total is simply the sum of the charges on the three capacitors: . For this parallel connection the potential difference is the same across each capacitor (the upper plates are all at the same potential and the potential of the lower plates differ from this by ). The charges on the capacitor must then be , , . We can therefore immediately write , which says simply that we add the capacitances of capacitors connected in parallel. In most practical cases, combinations of capacitors can be consolidated into groups of capacitors that are connected either in series or in parallel, which are then reduced step-by-step to find the single equivalent capacitance. It is important to be systematic as you work through a problem involving both series and parallel connections. Consider the following examples. EXAMPLE 4-5: Find the equivalent capacitance of the combination of capacitors shown in the figure at right, where , , and Solution: . In this problem we first consolidate and into an equivalent capacitor to obtain the simplified circuit at right. Now we have and in series, which gives the following relation for the equivalent capacitance : . Inserting the values given for EXAMPLE 4-6: A potential difference of an capacitor ( , , and , we obtain is applied to a . capacitor ( ) and ) connected in series. (a) Determine the charge and the potential difference for each capacitor. (b) The charged capacitors are disconnected and then reconnected with their positive plates together and negative plates together, no external voltage being applied. What are the charge and the potential difference for each capacitor after this procedure? (c) The charged capacitors in part (a) are reconnected with plates of opposite sign together. What are the charge and the potential difference for each? Solution: (a) Conservation of charge on the two middle plates in the diagram at right lets us conclude immediately that the two capacitors carry equal charges. After calculating the equivalent capacitance of , you should be able to show that , , . (b) The important part of the problem is to be sure that you can picture what is going on. For this purpose you may wish to sketch diagrams illustrating the successive steps of the problem. Diagrams (i) and (ii) show the capacitors disconnected and reconnected as described. Conservation of charge requires that the total charge on the upper plates in diagram (ii) is , and that the charge on the lower plates is the negative of this. We can thus draw diagram (iii) showing an equivalent capacitance , with charge on it and potential difference it. across Having calculated charges and , we can go back to diagram (ii) and determine the on and : , . Sometimes you may find that pursuing a solution in terms of the algebraic quantities gives a rather complicated expression, and it may be easier to compute intermediate numbers (such as in this case). Still, it is usually worthwhile to try writing out an algebraic expression before you give up and plug in numbers along the way. In this case, the direct method yields and These equations are not too complicated to evaluate in terms of the original variables. The potential across both capacitors is then (c) The sequence of steps in this case is similar to that in (b). The difference is that now we have . Practice Exercise 4-2 Write your solutions to the following problems in your notebook. 1. In the circuit diagrammed at right, the battery provides a constant potential difference of and and are initially uncharged. Switch the capacitor is closed, charging . Then is opened, disconnecting the battery from the circuit. Following this is closed. The value of the potential difference is then measured to be . Determine the capacitance of across . 2. In the circuit at right, the values of the capacitors are , , , and . (a) What is the equivalent capacitance between terminals and ? (b) What is the potential difference across terminals if a -volt battery is connected to and ? Check your answers with the Module 2 Answer Key and review Section 4-2 as necessary before going on. 4-3: Energy Storage and Energy Density OBJECTIVE: Determine the energy stored in a capacitor or combination of capacitors, and compute the energy stored per unit volume in a region where an electric field exists. Reading Assignment Study in your textbook Chapter 24, Section 3. Commentary The concept of electrical energy “stored” in empty space may seem a bit strange to you at first, but it stems rather naturally from our picture of an electric field that permeates the space surrounding any charged object. The plates in a charged capacitor exert forces on each other due to their opposite charges and this electrostatic force between the plates has the potential to do work on either the plates or the charges themselves. The amount of work that can be done by a capacitor (we will explore later how electrical work is measured in circuits) is proportional to the charge on the capacitor times the potential difference across it. Each electron (quantum of charge) that could be released from one plate of the capacitor would gain of kinetic energy per volt of potential change as it crossed the gap distance ( ) to the oppositely charged plate. The transfer of an increment of charge changes the potential, though, so we must integrate to find the total work that can be done by a capacitance carrying charge : (2) We choose, for the present, to think of as potential energy stored in the electric field as opposed to potential energy due to the charge separation on the capacitor (the two views are equivalent in our model). The energy per unit volume stored in a (vector) electric field can be characterized by a (scalar) field- the energy density - that is proportional to . For fields in vacuum, energy density is given by . As we will see in this next section, this result can be further generalized to electric fields in other media by adding a correction to the permittivity constant . Still, it depends on a detailed knowledge of , which is generally available only for capacitors of simple geometry (e.g., parallel-plate or radial shells). If we are interested merely in the total energy stored in a capacitor with charge and potential difference , we may use any of the relationships in Eq. (2). Study Sample Problems 24.7, 24.8, and 24.9 in the text and the following additional example. EXAMPLE 4-7: Two capacitors, and initially by being connected to a , are each charged battery. Then the two capacitors are connected together. What is the total electric energy stored in the capacitors, if they are connected such that (a) plates of like charge are connected together? (b) Plates of opposite charge are connected together? (c) Account for the lost energy in part (b). Solution: The first step here, as in previous problems, is to make sure you have a clear picture of the situation- drawing a diagram is recommended. Since , we have . We note that the stored energy after the initial charging is (a) When plates of like charge are connected together, charge is confined to the other pair of plates. Since the like-charged plates are already at the same potential , no charge is redistributed so that the potential across and energy stored in the capacitors remain the same as before they were connected: . In fact, the situation would be identical if the capacitors had been charged in parallel in the first place. (b) When plates of opposite charge are connected, we may visualize charge on the upper plates in the diagram and charge on the lower plates. The equivalent and the potential difference is . Therefore, the stored energy is capacitance is . (c) The energy that “disappears” when plates of unlike potential difference are connected goes into heating the wires and into electromagnetic radiation as the charges redistribute themselves through a flow of current from one plate to the other. Practice Exercise 4-3 Write your solutions to the following problems in your notebook. 1. How much energy is required to charge a metal sphere potential of in diameter to a ? 2. Refer to Example 4-5 in this syllabus. How much energy is stored in the capacitors if volts? 3. A to capacitor ( ) is charged to while an identical capacitor ( ) is charged . The oppositely charged plates are then connected together. Calculate the stored energy before and after the connection is made and the energy dissipated by the system. Check your answer with the Module 2 Answer Key and review Section 4-3 as necessary before going on to the next section. 4-4: Dielectrics OBJECTIVE: Describe the effect on a capacitor’s capacitance, voltage, charge, and stored energy, as well as the electric field in the capacitor, if the space between the conductors of the capacitor contains dielectric material; describe qualitatively the distribution of polarization charges that accounts for these effects. Reading Assignment Study in your textbook Chapter 24, Sections 4, 5, and 6. Commentary Any insulating material can be characterized by a dimensionless number- its dielectric constant, - which is a measure of the extent to which negative and positive charges in the material will separate in the presence of an electric field. Charge separation in a non-conducting material (dielectric) does not involve the movement of “free” charges, but rather the alignment of molecular dipoles within the material to produce a surfacebound polarization charge (Figure 24.18 in the text). The dielectric constant of a material is sometimes referred to as its relative permittivity; when we multiply by the permittivity of a vacuum, we obtain the absolute permittivity of the material , . Dielectric constants of selected materials are listed in the left column of Table 24.2 in the text. The dielectric constant enables us to generalize Gauss’s law so that it applies to charge distributions and resultant electric fields in media other than a vacuum. As you study the derivation of this generalized form in Section 24.6 of the text, make sure that you understand the significance of , the effective charge that appears on the surface of the dielectric. The essential thing to recognize is that all matter consists of charged particles, and that in the presence of an electric field these charges will tend to redistribute themselves (to a greater or lesser extent, depending on the molecular structure of the material) such that the field strength is reduced. Thus, the field strength resulting from a particular distribution of free charges is greatest in vacuum and can only be less in any material medium; the dielectric constant of the medium tells how much less. When you see a problem involving the insertion of a dielectric slab between the plates of a parallel-plate capacitor, be sure to notice whether or not the capacitor remains connected to a battery. If a battery is connected, you can assume that it will maintain a constant potential difference between the capacitor plates so that the free charge on the plates must increase as the dielectric is inserted. If the capacitor has been disconnected from the charging battery, the charge on the plates remains fixed and the potential between them must decrease as the dielectric is inserted. Study Sample Problems 24.10, 24.11, and 24.12 in the text and the following example. EXAMPLE 4-8: A parallel-plate capacitor is half filled with an insulating material of dielectric constant , as suggested by the diagram at right; a vacuum exists in the remaining space. What is the capacitance in terms of , , and ? Solution: We can solve this problem by treating the capacitor as two capacitors in parallel, each having an area of , with one capacitor filled with dielectric and the other in vacuum. The capacitance of the dielectricfilled plates is then and that of the plates in vacuum is so that the total capacitance is Notice that this reduces to the expected result in the limiting case where . Practice Exercise 4-4 Write your solutions to the following problems in your notebook. 1. A Geiger counter is made of two long, concentric metal cylinders with a gas of dielectric constant between them. Neglecting edge effects, use Gauss’s law to calculate the capacitance of the configuration. The center rod has radius , the surrounding tube radius , and the length of the cylinders is . 2. A parallel-plate capacitor of plate area and separation is charged by a battery to potential , and is then disconnected from the battery. (a) Give expressions for the energy stored in, and charge on, the capacitor. (b) A slab of dielectric with constant is then inserted into the capacitor, completely filling the space between the plates. Determine the capacitance, charge, potential difference, energy stored, and electric field in the capacitor. Explain quantitatively what happens to the “lost” energy, and describe the distribution of polarization charge in the dielectric. Check your answers with the Module 2 Answer Key and review Section 4-4 as necessary before doing the self-check test. To see if you have achieved the objectives for Lesson 4, try to solve the problems in Self-Check Test 4 without using any reference materials. Self-Check Test 4 Write your solutions to the following problems in your notebook. 1. Derive the expression for the capacitance of a capacitor that consists of concentric cylindrical conducting shells of radius and ( ) and of length . 2. A capacitor is charged to and an capacitor is charged to . They are then connected in parallel (positive plate of one to the negative plate of the other) with an initially uncharged capacitor. Determine the energy stored in the final configuration. 3. A parallel-plate capacitor with plate separation (a) If a dielectric slab of thickness has capacitance . and dielectric constant is inserted between the plates and parallel to them, determine the ratio of the capacitance with the dielectric in place to the capacitance without the dielectric. (b) Make a sketch showing the location of free and polarization charges (with their signs) when the capacitor is charged with the dielectric in place. Check your answers with the Module 2 Answer Key and review Lesson 4 as necessary. Assignment 4 When you have demonstrated mastery of the content of this lesson, log into the Mastering Physics website and work Assignment 4.