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Capacitance and Dielectrics Chapter 24 Capacitance and Dielectrics PowerPoint® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman Lectures by James Pazun Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Goals for Chapter 24 • To consider capacitors and capacitance • To study the use of capacitors in series and capacitors in parallel • To determine the energy in a capacitor • To examine dielectrics and see how different dielectrics lead to differences in capacitance Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley How to Accomplish these goals: Read the chapter Study this PowerPoint Presentation Do the homework: 11, 13, 15, 39, 41, 45, 71 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Introduction • When flash devices made the “big switch” from bulbs and flashcubes to early designs of electronic flash devices, you could use a camera and actually hear a high-pitched whine as the “flash charged up” for your next photo opportunity. • The person in the picture must have done something worthy of a picture. Just think of all those electrons moving on camera flash capacitors! Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Keep charges apart and you get capacitance Any two charges insulated from each other form a capacitor. When we say that a capacitor has a charge Q or that charge Q is stored in the capacitor, we mean that the conductor at higher potential has charge +Q and at lower potential has charge –Q. When the capacitor is fully charged the potential difference across it is the same as the vab that charged it. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Capacitance If we double the magnitude of Q, it stands to reason that the electric field between the plates doubles. This means that the potential difference also doubles. Even though these values change, the ratio of charge to potential difference doesn’t. The ratio of charge to potential difference is called the capacitance of the capacitor. C=Q/Vab The SI unit of capacitance is the farad. One farad is one coulomb per volt. Capacitance is a measure of the ability of a capacitor to store energy. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Calculating Capacitance-Capacitors in Vacuum The simplest kind of capacitor is the parallel plate capacitor, see 24.2. From example 21.13 we found that the electric field between parallel plates is: σ E = ε o We can rewrite sigma as the total charge on one plate divided by the area of the plate: σ Q E == ε ε A oo 1 Qd The field is uniform, and the separation is d, so:VEd== ab ε A QA o We can state: C ==ε o Vdab The capacitance only depends on the geometry of the conductor! Be sure to read page 911 for “good stuph.” Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley The unit of capacitance, the farad, is very large • Commercial capacitors for home electronics are often cylindrical, from the size of a grain of rice to that of a large cigar. • Capacitors like those mentioned above and pictured at right are microfarad capacitors. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Some examples of flat, cylindrical, and spherical capacitors • See just how large a 1 F capacitor would be. Refer to Example 24.1. • Refer to Example 24.2 to calculate properties of a parallel-plate capacitor. • Follow Example 24.3 and Figure 24.5 to consider a spherical capacitor. • Follow Example 24.4 and Figure 24.6 to consider a cylindrical capacitor. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 24.1 Summary and Homework A capacitor is any pair of conductors separated by an insulating material. When the capacitor is charged, there are charges of equal magnitude Q and opposite sign on the two conductors, and the potential Vab of the positively charged conductor with respect to the negatively charged conductor is proportional to Q. The capacitance C is defined as the ratio of Q to Vab. The SI unit of capacitance is the farad (F): 1F = 1C/V. A parallel-plate capacitor consists of two parallel conducting plates, each with area A separated by a distance d. If they are separated by vacuum, the capacitance depends only on d and A. For other geometries, the capacitance can be found by using the definition C=Q/Vab Homework: 11 and 13 Read 914 to 922 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Capacitors may be connected one or many at a time When the series capacitors are connected across a potential difference of Vab, C1 acquires a charge of Q. This pulls a charge of –Q to the opposite plate of C1. This had to come from somewhere, in this case the charge came from C2. The induced Q on C2 pulls a –Q from b. This is why the charge on capacitor plates in series is the same. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Capacitors may be connected one or many at a time We can write the potentials between a & c, c & b and a & b as follows: Q Q VVac ==1 VVcb ==2 C1 C2 ⎛⎞11 VVVVQab ==+12 =⎜⎟ + ⎝⎠CC12 V 11 And so: =+ QC12 C Since C = Q/V, 1/C = V/Q We can replace the capacitors in series with one equivalent capacitor, that has the same capacitance as the combined capacitors: 1111 = +++… CCCCeq 123 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley CAUTION The reciprocal of the equivalent capacitance of a series combination equals the sum of the reciprocals of the individual capacitances. The magnitude of charge is the same on all plates of all the capacitors in a series combination; however, the potential differences of the individual capacitors are not the same unless their individual capacitances are the same. The potential differences of the individual capacitors add to give the total potential difference across the series combination: Vtotal = V1 +V2 + V3 + . Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Capacitors may be connected one or many at a time For parallel wired capacitors, the upper plates are connected together to form an equipotential surface, as are the bottom plates. Hence for parallel capacitors: the potential difference for all capacitors is the same. The potential difference is V. The charges Q1 and Q2 are not necessarily equal. They depend on the capacitances C1 and C2. Q1 = C1V and Q2 = C2V Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Capacitors may be connected one or many at a time If we replace the two capacitors with one equivalent capacitor, the charge on that capacitor is: Q = Q1 + Q2 = (C1 + C2)V Or: Q/V = C1 + C2 Since Ceq = Q/V, Ceq = C1 + C2 + C3 + . Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley CAUTION The equivalent capacitance of a parallel combination equals the sum of the individual capacitances. The potential differences are the same for all the capacitors in a parallel combination; however, the charges on individual capacitors are not the same unless their individual capacitances are the same. The charges on the individual capacitors add to give the total charge on the parallel combination: Qtotal = Q1 + Q2 + Q3 + . Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Capacitance Trends: Series Parallel Vtotal = V1 + V2 + V3 Vtotal = V1 = V2 = V3 Qtotal = Q1 = Q2 = Q3 Qtotal = Q1 + Q2 + Q3 1/Ceq = 1/C1 + 1/C2 + 1/C3 Ceq = C1 + C2 + C3 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Calculations regarding capacitance • Refer to Problem-Solving Strategy 24.1. • Follow Example 24.5. • Follow Example 24.6. The problem is illustrated by Figure 24.10 below. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 24.2 Summary and Homework When capacitors are connected in series, the reciprocal of the equivalent capacitance Ceq equals the sum of the reciprocals of the individual capacitances. When capacitors are connected in parallel, the equivalent capacitance equals the sum of the individual capacitances. On page 936: 15 Read 922 to 928 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley The Z Machine—capacitors storing large amounts of energy • This large array of capacitors in parallel can store huge amounts of energy. When directed at a target, the discharge of such a device can generate temperatures on the order of 109K! Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy Storage in Capacitors The energy stored in a capacitor is equal to the amount of work required to charge it. We can calculate the potential energy U of a charged capacitor by calculating the amount of work required to charge it. Suppose the capacitor is already charged. It contains a charge Q and potential difference V. (V = Q/C) Let q and v be the charge and potential difference at any intermediate stage while charging. (v = q/C) An infinitesimal part of the work to move a part of the charge to the capacitor is: dW = vdq = qdq/C The total work: W 1 Q Q2 WdWqdq==∫∫ = 00CC2 Since Q=CV: 2 Q 11 UCVQV==2 = 22C 2 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Capacitor Energy Storage Q2 11 UCVQV==2 = 22C 2 Read Pages 918 and 919 for a further discussion about the above relationships. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Electric-Field Energy Electric-Field Energy is the energy that is stored in the field between the plates.
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