Capacitors. February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

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1 Capacitors February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

2 Review! The electric potential energy stored in a capacitor is given by! The field energy density stored in a parallel plate capacitor is given by! The field energy density in general is U u = 1 2 CV 2 1 V ε 2 d = 0 2 u = 1 ε 0E 2 2 September 17, 2008 Physics for Scientists & Engineers 2, Lecture 14 2

3 Capacitors with Dielectrics! Placing a dielectric between the plates of a capacitor increases the capacitance of the capacitor by a numerical factor called the dielectric constant, κ! We can express the capacitance of a capacitor with a dielectric with dielectric constant κ between the plates as C =κc air! Where C air is the capacitance of the capacitor without the dielectric! Placing the dielectric between the plates of the capacitor has the effect of lowering the electric field between the plates and allowing more charge to be stored in the capacitor C = q ΔV February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 3

4 Parallel Plate Capacitor with Dielectric! Placing a dielectric between the plates of a parallel plate capacitor modifies the electric field as E = E air κ =! ε 0 is the electric permittivity of free space! ε is the electric permittivity of the dielectric material ε =κε 0 q κε 0 A = q εa! Note that the replacement of ε 0 by ε is all that is needed to generalize our expressions for the capacitance! The potential difference across a parallel plate capacitor is ΔV = Ed = qd κε 0 A! The capacitance is then C = q ΔV = κε A 0 d February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 4

5 Dielectric Strength! The dielectric strength of a material measures the ability of that material to withstand potential difference! If the electric field strength in the dielectric exceeds the dielectric strength, the dielectric will break down and begin to conduct charge between the plates via a spark, which usually destroys the capacitor! A useful capacitor must contain a dielectric that not only provides a given capacitance but also enables the device to hold the required potential difference without breaking down! Capacitors are usually specified in terms of their capacitance and by the maximum potential difference that they are designed to handle February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 5

6 Dielectric Constant! The dielectric constant of vacuum is defined to be 1! The dielectric constant of air is close to 1 and we will use the dielectric constant of air as 1 in our problems! The dielectric constants of common materials are listed below (more are listed in the book in Table 24.1) February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 6

7 Microscopic Perspective on Dielectrics! Let s consider what happens at the atomic and molecular level when a dielectric is placed in an electric field! A polar dielectric material is composed of molecules that have a permanent electric dipole moment! A nonpolar dielectric material is composed of atoms or molecules that have no inherent electric dipole moment Dipole moment is induced by external electric field February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 7

8 Microscopic Perspective on Dielectrics! In both polar and non-polar dielectrics, the fields resulting from aligned electric dipole moments tend to partially cancel the original electric field + -! The resulting electric field inside the capacitor then is the original field minus the induced field E r = E E d February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 8

9 Example I! A parallel plate capacitor whose capacitance C is 13.5pF is charged by a battery to a potential difference of V =12.5V between its plates. The battery is now disconnected and material with κ=6.5 is slipped between the plates. (a) What is the potential energy before the material is inserted? (b) What is U after the material has been inserted? (a) (b) Key Idea: Because the battery is disconnected, the charge on the capacitor cannot change! February 5, 2014 Physics for Scientists&Engineers 2 10

10 Example II! A parallel plate capacitor whose capacitance C is 13.5pF is charged by a battery to a potential difference of V =12.5V between its plates. The battery is now disconnected and material with κ=6.5 is slipped between the plates. (a) What is the potential energy before the material is inserted? (b) What is U after the material has been inserted? (b) Key Idea: Because the battery is disconnected, the charge on the capacitor cannot change, but the capacitance does change (C--> κc)! February 5, 2014 Physics for Scientists&Engineers 2 11

11 Example III! A parallel plate capacitor whose capacitance C is 13.5pF is charged by a battery to a potential difference of V =12.5V between its plates. The battery is now disconnected and material with κ=6.5 is slipped between the plates. (b) What is U after the material has been inserted? The potential energy decreased by a factor κ. The missing energy, in principle, would be apparent to the person inserting the material. The capacitor would exert a tiny tug on the material and would do work on it, in amount February 5, 2014 Physics for Scientists&Engineers 2 12

12 Capacitance of a Coaxial Cable! Coaxial cables are used to transport signals between devices with minimum interference! A 20.0 m long coaxial cable is composed of a conductor and a coaxial conducting shield around the conductor! The space between the conductor and the shield is filled with polystyrene! The radius of the conductor is mm and the radius of the shield is 2.00 mm! PROBLEM! What is the capacitance of the coaxial cable? February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 14

13 Capacitance of a Coaxial Cable SOLUTION! We can think of the coaxial cable as a cylinder! The dielectric constant of polystyrene is 2.6! We can treat the coaxial cable as a cylindrical capacitor with r 1 = mm and r 2 = 2.00 mm, filled with a dielectric with κ = 2.6! The capacitance of the coaxial cable is C =κ 2πε L 0 ln r 2 r 1 ( ) 2π F/m = 2.6 C = F=1.39 nf ( ) ( ) 20.0 m ln m m February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 15

14 Capacitor Partially Filled with a Dielectric PROBLEM:! A parallel plate capacitor is constructed of two square conducting plates with side length L = 10.0 cm.! The distance between the plates is d = cm.! A dielectric with dielectric constant κ =15.0 and thickness cm is inserted between the plates.! The dielectric is L = 10.0 cm wide and L/2 = 5.00 cm long.! What is the capacitance of this capacitor? February 5, 2014 Chapter 24 17

15 Capacitor Partially Filled with a Dielectric SOLUTION:! We have a capacitor partially filled with a dielectric.! We can treat this capacitor as two capacitors in parallel.! One capacitor is a parallel plate capacitor with plate area A = L(L/2) and air between the plates.! The second capacitor is a parallel plate capacitor with plate area A = L(L/2) and a dielectric between the plates.! Sketch Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. February 5, 2014 Chapter 24 18

16 Capacitor Partially Filled with a Research Dielectric! The capacitance of a parallel place capacitor is: C 1 = ε A 0 d! If a dielectric is placed between the plates we have: C 2 = κ ε A 0 d! The capacitance of two capacitors in parallel is: C 12 = C 1 + C 2 Simplify! Putting our equations together gives us: C 12 = C 1 = ε A 0 d +κ ε A 0 d = ( κ +1 ) ε A 0 d February 5, 2014 Chapter 24 19

17 Capacitor Partially Filled with a Dielectric! The area of the plates for each capacitor is: A = L( L / 2)= L 2 / 2! Putting our expressions together gives us: C 12 = κ +1! Calculate ( ) ( ) ε 0 L 2 / 2! Putting in our numerical values: d ( = κ +1 )ε 0 L 2 2d ( )( F/m) ( m) 2 2( m) C 12 = = F February 5, 2014 Chapter 24 20

18 Capacitor Partially Filled with a Round C 12 = F = 283 pf! Double-check Dielectric! To double-check our answer, we calculate the capacitance of the capacitor without any dielectric: ( ) 2 ( ) ( ) m C 1 = F/m m! Calculate the capacitance of the capacitor with dielectric: C 1 = ( 15)35.4 pf = 531 pf = 35.4 pf! Our result for the partially filled capacitor is half of the sum of these two results, so it seems reasonable. February 5, 2014 Chapter 24 21

19 Supercapacitor / Ultracapacitor! Supercapacitors (ultracapacitors) are made using a material with a very large surface area between the capacitor plates! Two layers of activated charcoal are given opposite charge and are separated by an insulating material! This produces a capacitor with capacitance millions of times larger than ordinary capacitors! However, the potential difference can only be 2 to 3 V February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 22

Chapter 7: Polarization

Chapter 7: Polarization Joaquín Bernal Méndez Group 4 1 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces