extra3 - Chapter 5 Capacitance and Dielectrics 5.1 The...

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Unformatted text preview: Chapter 5 Capacitance and Dielectrics 5.1 The Important Stuff 5.1.1 Capacitance Electrical energy can be stored by putting opposite charges q on a pair of isolated con- ductors. Being conductors, the respective surfaces of these two objects are all at the same potential so that it makes sense to speak of a potential difference V between the two conduc- tors , though one should really write V for this. (Also, we will usually just talk about the charge q of the conductor pair though we really mean q .) Such a device is called a capacitor . The general case is shown in Fig. 5.1(a). A particular geometry known as the parallel plate capacitor is shown in Fig. 5.1(b). It so happens that if we dont change the configuration of the two conductors, the charge q is proportional to the potential difference V . The proportionality constant C is called the capacitance of the device. Thus: q = CV (5.1) The SI unit of capacitance is then 1 C V , a combination which is called the farad 1 . Thus: 1farad = 1F = 1 C V (5.2) The permittivity constant can be expressed in terms of this new unit as: epsilon1 = 8 . 85 10- 12 C 2 N m 2 = 8 . 85 10- 12 F m (5.3) 5.1.2 Calculating Capacitance For various simple geometries for the pair of conductors we can find expressions for the capacitance. Parallel-Plate Capacitor 1 Named in honor of the. . . uh. . . Austrian physicist Jim Farad (16021796) who did some electrical exper- iments in. . . um. . . Berlin. Thats it, Berlin. 71 72 CHAPTER 5. CAPACITANCE AND DIELECTRICS MT43MT113 MT45MT113 MT43MT113 MT45MT113 MT40MT97MT41 MT40MT98MT41 Figure 5.1: (a) Two isolated conductors carrying charges q : A capacitor! (b) A more common configu- ration of conductors for a capacitor: Two isolated parallel conducting sheets of area A , separated by (small) distance d . The most common geometry we encounter is one where the two conductors are parallel plates (as in Fig. 5.1(b), with the stipulation that the dimensions of the plates are large compared to their separation to minimize the fringing effect. For a parallel-plate capacitor with plates of area A separated by distance d , the capaci- tance is given by C = epsilon1 A d (5.4) Cylindrical Capacitor In this geometry there are two coaxial cylinders where the radius of the inner conductor is a and the inner radius of the outer conductor is b . The length of the cylinders is L ; we stipulate that L is large compared to b . For this geometry the capacitance is given by C = 2 epsilon1 L ln( b/a ) (5.5) Spherical Capacitor In this geometry there are two concentric spheres where the radius of the inner sphere is a and the inner radius of the outer sphere is b . For this geometry the capacitance is given by: C = 4 epsilon1 ab b- a (5.6) 5.1.3 Capacitors in Parallel and in Series Parallel Combination: Fig. 5.2 shows a configuration where three capacitors are com- bined in parallel across the terminals of a battery. The battery gives a constant potential 5.1. THE IMPORTANT STUFF5....
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extra3 - Chapter 5 Capacitance and Dielectrics 5.1 The...

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