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.

• 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 + . . .

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. Lets find the energy density between the plates of a parallel plate capacitor with area A and separation d: The stored potential is ½ CV2 and the volume between the plates is Ad. The energy density (u) is: 1 CV 2 u = 2 Ad We can rewrite the capacitance from eq 24.2 and V = Ed 1 to get: uE= ε 2 2 o This happens to be true for any electric field configuration in a vacuum.

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley CAUTION It’s a common misconception that electric field energy is a new kind of energy, different from the electric potential energy described before. This is not the case; it is simply a different way of interpreting electric potential energy. We can regard the energy of a given system of charges as being a shared property of all the charges, or we can think of the energy as being a property of ether electric field that the charges create. Either interpretation leads to the same value of the potential energy.

Try example 24.8

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 24.3 Summary and Homework The energy U required to charge a capacitor C to a potential difference V and a charge Q is equal to the energy stored in the capacitor. This energy can be thought of as residing in the electric field between the conductors; the energy density u (energy per unit volume) is proportional to the square of the electric-field magnitude.

On page 936: 39

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Dielectrics change the potential difference Adding dielectrics to capacitors allows for large capacitances in a small volume. It also increases the maximum potential difference, protecting against dielectric breakdown (prevents static discharge). The most relevant result to introducing dielectrics into capacitors for us is that it decreases the potential across a capacitor.

The capacitance without a dielectric is given by Co = Q/Vo, with dielectric C = Q/V. The ratio of C to Co is called the dielectric constant of the material:

K = C/Co

Q = CoVo = CV

We can say: C/Co = Vo/V or V = Vo/K When a dielectric is present the potential is reduced by a factor K. Table 24.1 shows the dielectric constants of many common materials. YOUYOU SHOULDSHOULD MEMORIZEMEMORIZE THISTHIS TABLE!!!TABLE!!! Just kidding. No dielectric is a perfect insulator, there is always leakage through the dielectric, but we ignore this effect in our calculations.

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley When you insert a dielectric . . . Because the potential is decreased by K, the electric field must also decrease by the same factor:

E = Eo/K (24.14) Since the field is related to the surface charge density, sigma must decrease as well. Since Q doesn’t change, how can sigma change? The charged plates induce a surface charge on the dielectric like that in figure 24.13b. What is the net charge on the dielectric material?

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Dielectric Capacitor Equations Lets call the surface charge density on the plates σ and on the dielectric σi. The net surface charge on the left side of figure 24.13b is (σ - σi). With and without dielectrics the electric field is: σ σ −σ E = E = i o ε o ε Using these in oequation 24.14: o ⎛⎞1 σσi =−⎜⎟1 ⎝⎠K

The product Kεo is called the permittivity of the dielectric, denoted by ε = Kεo The electric field can be expressed as: E = σ/ε

In empty space (vacuum) K = 1, ε = εo and the equations reduce to there previous form.

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Examples to consider, capacitors with and without dielectrics • Refer to Problem-Solving Strategy 24.2. • Follow Example 24.10 to compare values with and without a dielectric. • Follow Example 24.11 to compare energy storage with and without a dielectric. Figure 24.14 illustrates the example.

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Dielectric breakdown • A very strong electrical field can exceed the strength of the dielectric to contain it. Table 24.2 at the bottom of the page lists some limits.

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 24.4 Summary and Homework When the space between the conductors is filled with a dielectric material, the capacitance increases by a factor K called the dielectric constant of the material. The quantity ε = Kεo is called the permittivity of the dielectric. For a fixed amount of charge on the capacitor plates, induced charges on the surface of the dielectric decrease the electric field and potential difference between the plates by the same factor K. The surface charge results from polarization, a microscopic rearrangement of charge in the dielectric. Under sufficiently strong fields, dielectrics become conductors, a situation called dielectric breakdown. The maximum field that a material can withstand without breakdown is called its dielectric strength. In a dielectric, the expression for the energy density is the same as in vacuum but with εo replaced by ε = Kεo.

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