Chapter 24 Capacitance, Dielectrics, Electric Energy Storage

Transcription

1 Chapter 24 Capacitance, Dielectrics, Electric Energy Storage Units of Chapter 24 Capacitors (1, 2, & 3) Determination of Capacitance (4 & 5) Capacitors in Series and Parallel (6 & 7) Electric Energy Storage (8-12, 14, & 15) Dielectrics (13) Molecular Description of Dielectrics 24.1 Capacitors A capacitor has the aility to store electric charge and widely used for: Storing charge foamera flashes and a ackup energy source foomputers. Protecting circuits y locking energy surges. Tuning Radio frequencies Memory of the inary code in RAM 1

2 24.1 Capacitors A capacitoonsists of two conductors that are close ut not touching Capacitors Parallel-plate capacitoonnected to attery. () is a circuit diagram (Note the symols used) Capacitors When a capacitor is connected to a attery, the charge on its plates is proportional to the voltage: The quantity C is called the capacitance. Unit of capacitance: the farad (F) 1 F = 1 C/V Q = C The capacitance of most capacitors are 1 pf - 1!F 2

3 HOW DO WE DESCRIBE AND APPLY THE CONCEPT OF ELECTRIC POTENTIAL? Calculating the Potential Difference,, in a Uniform Field (1) Because the field and the direction of motion are the same, cos θ = 1, (2) Because E is uniform, it can e pulled out of the integral. = E dl = E dl = Edlcosθ; a a ( ) = Edlcos 0 = E dl a!" # $# = E dl a a=0 = Ed 1 (3) The negative sign implies that the potential is decreasing as it moves a distance, d, in the electric field. a =d 24.2 Determination of Capacitance For a parallel-plate capacitor: Gauss s Law (Cylinder emedded in a plate)! E da = Q encl = σ A (Note: σ = Q and Q = σa ) A E2 A = σ A E = σ for each plate 2 E net = E = E 1 + E 2 = σ + σ = σ = Q 2 2 A 24.1 Capacitors The capacitance does not depend on the voltage; it is a function of the geometry (size, shape, and relative position of the two conductors) of and material that separates the capacitor. For a parallel-plate capacitor: 3

4 24.2 Determination of Capacitance For a parallel-plate capacitor: The relation etween electric field and electric potential is given y = E dl a We can take the line integral along a path antiparallel to the field lines, from one plate to the other; then θ = 180! and cos180! = 1, so = V V a = E dlcos180! = + E dl a = Q a A dl = Qd a A This relates Q to, and from it we can get the capacitance C in terms of the geometry of the plates: C = Q = Q = ε A 0 Qd d A 24.2 Determination of Capacitance For a cylindrical capacitor: Gauss s Law (Cylinder around a long wire)! E da = EA side + EA top + EA ottom = Q encl EA side + EA " + EA top #$ % ottom &% = E 2πrL =0 =0 E( 2πrL) = Q E = ( ) = Q encl Q 2π rl = 1 Q 2π Lr 24.2 Determination of Capacitance For a cylindrical capacitor: To otain C=Q/, we need to determine the potential difference etween the cylinders,, in terms of Q. To otain in terms of Q, write the line integral from the outeylinder to the inner one (so > 0) along a radial line. [Note that E points outward ut dl points inward for ouhosen direction of integration; the angle etween E and dl is 180 and cos 180 = -1. Also, dl = -dr ecause dr increases outward. These minus signs cancel.] 4

5 24.2 Determination of Capacitance For a cylindrical capacitor: = E dl = Q a 2π L = Q 2π L ln C = Q Q = Q 2π L ln R a R R R a R R a dr r = Q 2π L ln = 2πε L 0 ln R a R R a R 24.2 Determination of Capacitance Recommended Practice: Solve for a Spherical Capacitor r C = 4πε a r o r a r Recommended Application: Example 24.1 on page 615 HOW DO WE DESCRIBE AND APPLY THE CONCEPT OF ELECTRIC POTENTIAL? Concentric Conducting Spherical Shells r E + + da sphere + A 1 + r 1 r a + + r A 2 r 3 A 3 At radius r 3 ( r > r ), E da = EA 3 cosθ = EA 3 = E 4πr 2! ( ) = Q encl Because the enclosed charge outside of a conductor is zero, E outside = 0 At radius r 2 ( r a < r < r ),! E da = EA 2 cosθ = EA 2 = E 4πr 2 Therefore, E etween = 1 Q 2 4π r 1 At radius r 1 ( r < r a ), E da = EA 1 cosθ = EA 2 = E 4πr 2! Because the enclosed charge inside of a conductor is zero, E inside = 0 ( ) = Q encl ( ) = Q encl 5

6 HOW DO WE DESCRIBE AND APPLY THE CONCEPT OF ELECTRIC POTENTIAL? Determination of V from E: Between Concentric Spheres (r a <r <r ) r + r a dr r a dr rd r E outside =0 E = 1 Q 4π r ; 2 V cd = E c d r d E dl = E dr! cosθ = E dr r d 1 Q V cd = 4π r dr = Q r d 1 r 2 c 4π r dr r 2 c V cd = Q 1 4π r r d = =1 r d = Q 1 4π r Q r V cd = c r d 4π r d at r a < < r d < r Note: If r a and r d r, then Q r = a r 4π r a r r d Q 1 1 4π r d Determination of V from E: Between Concentric Cylinders (r a <r<r ) E inside =0 At radius r a < r < r, d r d V E etween = λ cd = E dl = 1 2π c E dr! cosθ = r d =1 E dr r r d 1 λ V cd = E 2π r dr = λ r d 1 2π r dr V cd = λ ln 1 2πε 0 r r d = λ ln 1 2πε 0 r r d = λ ( ln( r d ) ln( )) 2π E outside =0 λ V cd = ln r d 2π at r a < < r d < r Note: If r a and r d r, then λ = ln r 2π r a 24.3 Capacitors in Series and Parallel Capacitors can e connected in two ways: In Parallel In Series 6

7 24.3 Capacitors in Parallel When a attery of voltage V is connected such that all of the left hand plates reach the same potential V a and all of the right hand plates reach the same potential V Capacitors in Parallel Each capacitor plate acquires a charge given y: Q 1 = C 1 V Q 2 = C 2 V The total charge Q that must leave the attery is then: Q = Q 1 + Q 2 = C 1 V + C 2 V Finding a single equivalent capacitor that will hold the same charge Q at the same voltage V gives: Q = C eq V 24.3 Capacitors in Parallel Finding a single equivalent capacitor that will hold the same charge Q at the same voltage V gives: Q = C eq V Comining the aove equation with the previous equation gives: C eq V = C 1 V + C 2 V = ( C 1 + C 2 )V or C eq = C 1 + C 2 7

8 24.3 Capacitors in Parallel The net effect of connecting capacitors in parallel increases the capacitance ecause we are increasing the area of the plates where the charge can accumulate: 24.3 Capacitors in Series When a attery of voltage V is connected to capacitors that are connected end to end. A charge +Q flows from the attery to one plate of C 1, and -Q flows to one plate of C 2. The region was originally neutral; so the net charge must still e zero Capacitors in Series The +Q on the left plate of C 1 attracts a charge of -Q on the opposite plate. Because region must have a zero net charge, there is thus a +Q on the left plate of C 2. 8

9 24.3 Capacitors in Series A single capacitor that could replace these two in series without affecting the circuit (Q and V stay the same) would have a capacitance C eq where: Q = C eq V V = Q C eq The total voltage V across the two capacitors in series must equal the sum of the voltages across each capacitor V = V 1 + V Capacitors in Series Because each capacitor plate acquires a charge given y: Q = C 1 V 1 V 1 = Q C 1 Q = C 2 V 2 V 2 = Q C Capacitors in Series Solving each for V and comining the previous equations gives: Q = Q + Q = Q C eq C 1 C 2 C 1 C 2 or 1 = C eq C 1 C 2 Note that the equivalent capacitance C eq is smaller than the smallest contriuting capacitance. 9

10 24.3 Capacitors in Series and Parallel Application: Determine the capacitance, C eq, of a single capacitor with the same effect as the 4 capacitors comined in series-parallel. (Let C 1 = C 2 = C 3 = C 4 = C.) 24.3 Capacitors in Series and Parallel Application (con t): Determine the capacitance, C eq, of a single capacitor with the same effect as the 4 capacitors comined in series-parallel. (Let C 1 = C 2 = C 3 = C 4 = C.) Determine the charge on each capacitor and potential difference across each if the capacitors were charged y a 12-ttery Capacitors in Series and in Parallel Capacitors in parallel have the same voltage across each one: 10

11 24.3 Capacitors in Series and in Parallel In this case, the total capacitance is the sum: [parallel] (24-3) 24.3 Capacitors in Series and in Parallel Capacitors in series have the same charge: 24.3 Capacitors in Series and in Parallel In this case, the reciprocals of the capacitances add to give the reciprocal of the equivalent capacitance: [series] (24-4) 11

12 24.4 Electric Energy Storage A charged capacitor stores electric energy; the energy stored is equal to the work done to charge the capacitor. The net effect of charging a capacitor is to remove charge from one plate and add it to the other plate. This is what the attery does when it is connected to a capacitor. A capacitor does not ecome charged instantly. It takes time. Initially, when a capacitor is uncharged, it requires no work to move the first it of charge over Electric Energy Storage When some charge is on each plate, it requires work to add more charge of the same sign ecause of electric repulsion. The more charge already on the plate, the more work is required to add additional charge Electric Energy Storage The work needed to add a small amount of charge dq, when a potential difference V is across the plates is dw = V dq. Since V=q/C at any moment where C is the capacitance, the work needed to store a total charge Q is W = Vdq = 1 Q qdq C = 1 Q C Q 0 12

13 24.4 Electric Energy Storage The energy stored in a capacitor is U = 1 Q 2 2 C When the capacitor C carries charges +Q and -Q on its two conductors. Since Q=CV (and V=Q/C), where V is the potential difference across the capacitor, we can also write U = 1 Q 2 2 C = 1 2 CV 2 = 1 2 QV 24.4 Electric Energy Storage It is useful to think of the energy stored in a capacitor as eing stored in the electric field etween the plates. Using U = 1 2 CV A 2, C = ε o, and V = Ed d for a parallel plate capacitor, solve the energy per volume as a function of the electric field: U Volume (E) = 24.4 Electric Energy Storage The energy density, defined as the energy per unit volume, is the same no matter the origin of the electric field: U energy density = u = Volume = 1 2 ε oe 2 The sudden discharge of electric energy can e harmful or fatal. Capacitors can retain their charge indefinitely even when disconnected from a voltage source e careful! 13

14 24.4 Storage of Electric Energy Heart defirillators use electric discharge to jump-start the heart, and can save lives Dielectrics A dielectric is an insulator that is placed etween two capacitor plates The Purpose of Dielectrics 1. They do not allow charge to flow etween them as easily as in air. The result: higher voltages can e applied without charge passing through the gap 2. They allow the plates to e placed closer together without touching. The result: the capacitance is increased ecause d is less. 14

15 24.5 The Purpose of Dielectrics 3. If the dielectric fills the space etween the two conductors, it increases the capacitance y a factor of K, the dielectric constant. C = KC 0 where C 0 is the capacitance when the space etween the two conductors of the capacitor is a vacuum and C is the capacitance when the space is filled a material whose dielectric constant is K. TABLE 24-1 Dielectrics Dielectric strength is the maximum field a dielectric can experience without reaking down. Note the similarity etween a vacuum and air Dielectrics Capacitance of a parallel-plate capacitor filled with dielectric: A C = Kε o d Because the quantity Kε o appears so often in formulas, a new quantity known as the permittivity of the material is defined as ε = Kε o 15

16 24.5 Dielectrics Therefore the capacitance of a parallel-plate capacitor ecomes C = ε A d The energy density stored in an electric field E in a dielectric is given y u = 1 2 Kε o E 2 = 1 2 εe2 Application #1: 24.5 Dielectrics A parallel-plate capacitor, filled with a dielectric with K = 2.2, is connected to a 12 V attery. After the capacitor is fully charged, the attery is disconnected. The plates have area A = 2.0 m 2, and are separated y d = 4.0 mm. Find the (a) capacitance, () charge on the capacitor, (c) electric field strength, and (d) energy stored in the capacitor. Application #2: 24.5 Dielectrics The dielectric from the previous parallel-plate capacitor is carefully removed, without changing the plated separation nor does any charge leave the capacitor. Find the new values of (a) capacitance, () the charge on the capacitor, (c) the electric field strength, and (d) the energy stored in the capacitor. 16

17 Molecular Description of Dielectrics The molecules in a dielectric tend to ecome oriented in a way that reduces the external field. Molecular Description of Dielectrics This means that the electric field within the dielectric is less than it would e in air, allowing more charge to e stored for the same potential. Partial Summary of Chapter 24 Capacitors in parallel: Capacitors in series: 17