Electricity and Magnetism Capacitors and Dielectrics

Transcription

1 Electricity and Magnetism Capacitors and Dielectrics Lana Sheridan De Anza College Oct 20, 2015

2 Last time circuits capacitors in series and parallel

3 Warm Up Question pictorial presentation of two pacitors connected in ries to a battery A circuit diagram showing the two capacitors connected A circuit diagram showing the equivalent capacitance of the Two capacitors of values 4.0 nf and 6.0 nf are connected in a in series to a battery circuit as shown: capacitors in series C 1 V 2 C 2 C 1 C 2 Q Q Q V V 1 V 2 V C eq C 1 C 2 V (A) 4.0 nf b c (B) 6.0 nf ividual (C) capacitances. 10 nf Statement (2) makes sense because we are essentially ing the areas of all the capacitor plates when they are connected with conwire, (D) and 2.4 capacitance nf of parallel plates is proportional to area (Eq. 26.3).

4 Warm Up Question pictorial presentation of two pacitors connected in ries to a battery A circuit diagram showing the two capacitors connected A circuit diagram showing the equivalent capacitance of the Two capacitors of values 4.0 nf and 6.0 nf are connected in a in series to a battery circuit as shown: capacitors in series C 1 V 2 C 2 C 1 C 2 Q Q Q V V 1 V 2 V C eq C 1 C 2 V (A) 4.0 nf b c (B) 6.0 nf ividual (C) capacitances. 10 nf Statement (2) makes sense because we are essentially ing the areas of all the capacitor plates when they are connected with conwire, (D) and 2.4 capacitance nf of parallel plates is proportional to area (Eq. 26.3).

5 Overview practice with capacitors in circuits energy stored in a capacitor dielectrics

6 Capacitors in Series and Parallel In general, for any number n of capacitors in series, we can always relate the effective capacitance of them all together to the individual capacitances by: 1 C eq = 1 C 1 1 C C n = n 1 C i i=1 The equivalent capacitance of capacitors in series is always less than the smallest capacitance in the series. And a reminder, in capacitors in parallel: C eq = C 1 C 2... C n = n i=1 C i

7 We first reduce the circuit to a single capacitor. More Practice with Multiple Capacitors What is the equivalent capacitance of this arrangement? A The equival parallel cap is larger. A C 1 = 12.0 µf V C 3 = 4.50 µf (a) C 2 = 5.30 µf B V C C (b)

8 itance More Practice When solving this type of problem, take an iterative approach. itance Identify sets of capacitors that are in parallel, then series, then parallel, etc. and at each step replace with the equivalent capacitance: and b for the re 26.9a. All ly and make e connected. llel connec- a a b a b b

9 y More can be Practice connected in series or pacitance for the combination, parallel When(c) solving either this way type because of problem, take an iterative approach. Identify sets of capacitors that are in parallel, then series, then parallel, etc. and at each step replace with the equivalent capacitance: the All ake ted. ecins 6.0 a b a b a b a 6.0 b a b c d Figure 26.9 (Example 26.3) To find the equivalent capacitance

10 More Practice We first reduce the circuit to a single capacitor. What is the equivalent capacitance of this arrangement: A The equival parallel cap is larger. A C 1 = 12.0 µf V C 3 = 4.50 µf (a) C 2 = 5.30 µf B V C C (b)

11 More Practice We first reduce the circuit to a single capacitor. What is the equivalent capacitance of this arrangement: A The equival parallel cap is larger. A C 1 = 12.0 µf V C 3 = 4.50 µf C 2 = 5.30 µf B V C C C eq = 3.57 µf. (a) (b)

12 Energy Stored in a Capacitor A charged capacitor has an electric field between the plates. This field can be thought of as storing potential energy. The energy stored in a capacitor with charge q and capacitance C is ( ) q 2 U = 1 2 C Since q = CV we can also write this as: U = 1 C ( V )2 2

13 Stored Energy Example Suppose a capacitor with a capacitance 12 pf is connected to a 9.0 V battery. What is the energy stored in the capacitor s electric field once the capacitor is fully charged?

14 Stored Energy Example Suppose a capacitor with a capacitance 12 pf is connected to a 9.0 V battery. What is the energy stored in the capacitor s electric field once the capacitor is fully charged? U E = J

15 Energy Density It is sometimes useful to be able to compare the energy stored in different charged capacitors by their stored energy per unit volume. We can link energy density to electric field strength. This will make concrete the assertion that energy is stored in the field. For a parallel plate capacitor, energy density u is: u = U Ad

16 Energy Density It is sometimes useful to be able to compare the energy stored in different charged capacitors by their stored energy per unit volume. We can link energy density to electric field strength. This will make concrete the assertion that energy is stored in the field. For a parallel plate capacitor, energy density u is: u = U Ad (Ad is the volume between the capacitor plates.)

17 Energy Density and Electric Field u E = U E Ad = C( V )2 2Ad

18 Energy Density and Electric Field u E = U E Ad = C( V )2 2Ad Replace C = ɛ 0A d : u E = ɛ 0A V 2 d 2Ad = ɛ ( 0 V 2 d ) 2

19 Energy Density and Electric Field u E = U E Ad = C( V )2 2Ad Replace C = ɛ 0A d : u E = ɛ 0A V 2 d 2Ad = ɛ ( 0 V 2 d ) 2 Lastly, remember V = Ed in a parallel plate capacitor, so: u E = 1 2 ɛ 0E 2

20 Dielectrics dielectric an insulating material that can affects the strength of an electric field passing through it Different materials have different dielectric constants, κ.

21 Dielectrics dielectric an insulating material that can affects the strength of an electric field passing through it Different materials have different dielectric constants, κ. κ tells us how the capacitance of a capacitor changes if the material between the plates is changed. For air κ 1. (It is 1 for a perfect vacuum.) κ is never less than 1. It can be very large > 100.

22 Dielectrics and Capacitance dielectric an insulating material that can affects the strength of an electric field passing through it The effect of sandwiching a dielectric in a capacitor is to change the capacitance: C κc κ is the dielectric constant.

23 Dielectric in a Capacitor CHAPTER CHAPTER 25 CAPACITANC 25 Capacitance C Capacitance κc B κ κ B Adding a dielectric increases the capacitance. V = a constant V = a constant (a) (a)

24 Effect of a Dielectric The most straightforward way of tracking quantities that will change when a dielectric is added is by replacing ɛ 0 in all equations with ɛ using this relation: ɛ = κɛ 0 (Or just think of the effect of the dielectric being ɛ 0 κɛ 0.) The electrical permittivity increases.

25 Dielectric in a Capacitor For a parallel plate capacitor with a dielectric, the capacitance is now: C = κɛ 0A d

26 Dielectric in a Capacitor If we add 670a dielectric CHAPTER while the capacitor 25 CAPACITANCE is connected to a battery: B κ B V = a constant (a) Fig (a) If the potential difference between the plates of a capacitor is maintained, as by battery B, the effect of a dielectric is to increase the charge on the plates. (b) If the charge on the capacitor plates is increase the cap tanate, can incre Another ef difference that the breakdown

27 Dielectric in a Capacitor If we add 670a dielectric CHAPTER while the capacitor 25 CAPACITANCE is connected to a battery: B κ B V = a constant (a) qfig. will increase (q (a) = If CV the ) potential difference U between will increase. the plates (U = of 1 a capacitor is maintained, as by battery B, CV 2 ) 2 the effect of a dielectric is to increase the charge on the plates. (b) If the charge on the capacitor plates is increase the cap tanate, can incre Another ef difference that the breakdown

28 Dielectric in a Capacitor If we add a dielectric while the capacitor is isolated so charge cannot leave the plates: B 0 VOLTS κ 0 VOLTS q = a constant (b) ncrease the capacitance of a capacitor, and some materials, such as strontium anate, can increase the capacitance by more than two orders of magnitude. Another effect of the introduction of a dielectric is to limit the poten ifference that can be applied between the plates to a certain value V max,ca he breakdown potential. If this value is substantially exceeded, the dielec

29 Dielectric in a Capacitor If we add a dielectric while the capacitor is isolated so charge cannot leave the plates: B 0 VOLTS κ 0 VOLTS q = a constant (b) ncrease the Vcapacitance will decrease. of (V a capacitor, = q C ) and some materials, such as strontium anate, can increase the capacitance by more than two orders of magnitude. Another U will effect decrease. of the (U introduction = q2 2C ) of a dielectric is to limit the poten ifference that can be applied between the plates to a certain value V max,ca he breakdown potential. If this value is substantially exceeded, the dielec

30 Effect of a Dielectric on Field Imagine again the isolated conductor: charge density σ is constant. B 0 VOLTS κ 0 VOLTS q = a constant (b) The electric field between the plates is E = increase the capacitance of a capacitor, and some σ ɛmaterials, 0 originally. such as strontium t tanate, can increase the capacitance by more than two orders of magnitude. Another With dielectric effect added: of the introduction E σ κɛ 0. of a dielectric is to limit the potenti difference that can be applied between the plates to a certain value V max,calle the breakdown The field strength potential. decreases! If this value E is E substantially exceeded, the dielectr κ material will break down and form a conducting path between the plates. Ever dielectric material has a characteristic dielectric strength, which is the maximu value of the electric field that it can tolerate without breakdown. A few suc

31 Effect of a Dielectric on Field Imagine again the isolated conductor: charge density σ is constant. B 0 VOLTS κ 0 VOLTS q = a constant (b) The electric field between the plates is E = increase the capacitance of a capacitor, and some σ ɛmaterials, 0 originally. such as strontium t tanate, can increase the capacitance by more than two orders of magnitude. Another With dielectric effect added: of the introduction E σ κɛ 0. of a dielectric is to limit the potenti difference that can be applied between the plates to a certain value V max,calle the breakdown The field strength potential. decreases! If this value E is E substantially exceeded, the dielectr κ material will break down and form a conducting path between the plates. Ever dielectric What material happens has to the a characteristic energy density dielectric u? strength, which is the maximu value of the electric field that it can tolerate without breakdown. A few suc

32 Effect of a Dielectric on Field What happens to the energy density? Was: u 0 = 1 2 ɛ 0E 2 0. u = 1 2 (κɛ 0) (E) 2

33 Effect of a Dielectric on Field What happens to the energy density? Was: u 0 = 1 2 ɛ 0E 2 0. Energy density decreases. u = 1 2 (κɛ 0) (E) 2 = 1 ( σ 2 (κɛ 0) κɛ 0 = 1 ( 1 2 ɛ 0κ = 1 κ u = u 0 κ κ 2 ) 2 ) E 2 0 ( ) 1 2 ɛ 0E0 2

34 Dielectrics and Electric Field Dielectrics effect the field around a charge E E κ For example, for a point charge q in free space: E 0 = k q r 2 = 1 4πɛ 0 q r 2 But in a dielectric, constant κ: E = 1 q 4π(κɛ 0 ) r 2 = E 0 κ

35 Dielectrics and Electric Field Dielectrics effect the field around a charge E E κ For example, for a point charge q in free space: E 0 = k q r 2 = 1 4πɛ 0 q r 2 But in a dielectric, constant κ: E = 1 q 4π(κɛ 0 ) r 2 = E 0 κ But how does this happen?

36 Dielectrics and Electric Field Dielectrics become polarized by the presence of an electric field. There are two types of dielectrics, the process is a little different in each: polar dielectrics nonpolar dielectrics

37 inuously jostling each other as a result of their random thermal ent Polar is not complete, Dielectrics but it becomes more complete as the maglied field is increased (or as the temperature, and thus the sed). The Thealignment external electric of the electric field partially dipoles aligns produces thean molecules electric of the opposite dielectric the applied with the field field. and is smaller in magnitude. ules c dipole andom e of an ) An roducthe n pret. p (a) Since the dielectric is an insulator, there are no free charges to move through the substance, but molecules can align. eg. distilled water (b) 1 Figures from Halliday, Resnick, Walker, 9th ed.

38 Nonpolar Dielectrics Nonpolar dielectrics are composed of molecules which are not polar. However, under the influence of a field, the distribution of the electrons in the molecules, or the shape of the molecule, is altered. Each molecule becomes slightly polarized. a S E b

39 Nonpolar Dielectrics CHAPTER 25 CAPACITANCE Nonpolar dielectrics are composed of molecules which (a) are not polar. 2. Nonpolar dielectrics. dipole moments, mo placed in an externa that this occurs beca slightly separating the Figure 25-15a shows applied. In Fig b, a are charged as shown. T tive and E negative charge 0 on one face of the slab charge on the opposite f (b) as a whole remains elect charge in any volume ele However, The initial under electric the influence field of a field, the distribution of the The applied field electrons inside this in thenonpolar molecules, or the shape of the molecule, is altered. aligns the atomic Each dielectric molecule slab becomes is zero. slightly polarized. dipole moments. E 0 = 0 eg. nitrogen, (a) benzene

40 Electric field inside the dielectric The field of the aligned atoms is opposite the The polarized dielectric contributes its own field, E. applied field. E E 0 (c) E' dielectric m shows a pa dielectric. W tions. Note dielectric by For the field E : 0 betw top plate wi the magnitu This reduces the electric field from the charged plates alone or E 0. Fig (a) A nonpolar dielectric slab. The circles represent the electrically In Fig. The resulting reduced field is E = E 0 neutral atoms within the slab. κ (b) An electric field is applied via charged capacitor face. between the Howev

41 E : da : 0 EA q, Guass s Law 0 with dielectrics (25-30) E 0 q κɛ 0 Φ E = q free (25-31) 0 A. or: 5-16b, with the dielectric in place,we can E da = q find the electric field plates (and within the dielectric) by using the free same Gaussian surer, now the surface encloses two types of κɛ charge: 0 It still encloses A te ithth a diserted. n the ed to both Gaussian surface E 0 (a) q q Gaussian surface κ The charge q free = q in the diagram. It is just the charge on the plates, the charge that is free to move. (b) E q q q' q'

42 Electric Displacement It is sometimes convenient to package the effect of the electric field together with the effect of the dielectric. For this, we introduce a new quantity, Electric Displacement. D = κɛ 0 E Gauss s law is very often written in terms of the electric displacement, rather than the electric field, if the field being studied is in a polarizable material.

43 plates of the capacitor are the wallboard and air. Uses of Dielectric Effects Capacitor plates Stud finder Wallboard Stud 1 Figures from Serway When & the Jewett, capacitor 9th ed. moves across commonly oil (Fig. 26. Often, a low voltage tact with an ions contai electrolyte, layer serves an electroly plate separ Electroly have a pola device. Wh rect. If the oxide layer

44 Uses of Dielectric Effects Computer keyboard: Key Movable plate Insulator Fixed plate B Figure 26.3 (Quick Quiz 26.2) One type of computer keyboard 25.6) is sm voltage of the batter more cha plates. Wh battery, th the charg result, the Q uick Qu as show the mo what ha in a wa

45 Summary practice with capacitors in circuits energy stored in a capacitor dielectrics Homework Halliday, Resnick, Walker: PREVIOUS: Ch 25, onward from page 675. Questions: 1, 3, 5; Problems: 1, 3, 5 NEW: Ch 25, Problems: 9, 11, 13, 19, 29, 31, 45