Chapter 24: Capacitance and Dielectrics When you compress/stretch a spring, we are storing potential energy This is the mechanical method to store energy It is also possible to store electric energy as electric potential energy using capacitors In this chapter: we will study electric devices (capacitors) used to store electric energy We Will Learn The Followings The nature of capacitors (how we can make them) and their ability to store energy There are various types of capacitors, we will define the physical quantity (capacitance) which is a measure of ability to store energy and learn how to calculate this quantity for various capacitor like the plate type cylindrical spherical Analysis of capacitors connected in a network (connected in parallel and series) There might be more than one capacitor connected in a circuit Calculation of energy stored in a capacitor (how to calculate the stored energy) We will drive an expression that gives the total amount of stored energy in a capacitor. Study of dielectrics in more details and how they used to make capacitors more effective We will learn how we can use the dielectrics to increase the efficiency of capacitors.
Capacitors and Capacitance Capacitors are the devices that store electrical energy as electric potential energy A capacitor can be made using two conductors insulated from each other Q = 0 (uncharged empty) no charge means the capacitors are not charged or empty Q 0 (charged) One can charge this capacitor by using a battery by applying a potential difference which will reveal +Q and Q on each conductor. This process is called charging the capacitor. The battery doesn t produce any charged. It just does move the charges from one conductor to another through chemical process until the potential difference becomes equal to potential difference along the battery. Capacitors in Circuit Diagrams The amount of charge on each conductor Q is proportional to potential difference between the conductors. This proportionality constant / ratio between Q and V ab is called capacitance:
Capacitance *** Q ~ V ab *** proportionality constant is called capacitance, C. When potential difference increases Q increases, but C remains the same. C is a CONSTANT that ONLY depends on the physical properties of the capacitor mostly the geometry C is a measure of ability to store energy. Greater C means that it can hold/store more charge and energy Unit for capacitance The SI unit of capacitance is called one farad (1 F), named after 19 th century English physicist Michael Faraday. Attention: Don t confuse C with C C is used for coulomb (unit for charge) but C (italic C) is for capacitance.
Calculating Capacitance: Capacitors in Vacuum There are various type of capacitors commonly used in applications Parallel plate Capacitors formed using two conducting parallel plates Spherical Capacitors formed using two conducting spherical shells Cylindrical Capacitors formed using two conducting cylindrical shells Calculation of Capacitance for these capacitors is simple. For a capacitor, we can calculate the capacitance by considering a charge Q then calculate the potential difference between them and then applying formula that describes the capacitance. Here, we will consider the capacitors in vacuum where the space between the conductors are empty. Later we will learn using the dielectrics to increase the capacitance (measure of efficiency in storing energy) of a capacitor.
Parallel Plate Capacitors Calculation What is the capacitance of a capacitor which is formed using two parallel plates? The separation between the plates is d and surface area for the plates is A. In order to determine the capacitance Consider plates are charged with +Q and Q Determine the potential between + and plates Apply capacitance formula C = Q/V ab If the d << size of the plates then the field is uniform directed from + to plate. From Gauss s Law: which can be expressed in terms of Q: Potential difference: yields to Now, Apply capacitance Formula: C = Q /V ab C Q Q A C 0 Ed Q A d d 0
Parallel Plate Capacitors Interpretation As mentioned before, capacitance of any capacitor only depends on physical properties of the plate and vacuum (that fills the space between the pates) C 0 A d Capacitance Increases with increasing A Decreases with increasing d Capacitance also depends on ε 0 (electric permittivity of free space) which is the nature of space between the plates. Later, we will see that we can use some dielectric materials to increase the capacitance instead of using vacuum. Let s now do a unit analysis for the expression that we have found. As you remember from Coulomb s law, unit for ε 0 : plug this into eq. Remember Joule/Coulomb = Volt So F = Coulomb/Volt is dimensionally correct Therefore, we can take the unit for ε 0 alternatively in capacitance calculations Note: Because ε 0 is very small, capacitors in industry are usually in microfarads or smaller, like picofarad 1pF = 10 12 F
Various Types of Commercial Capacitors Capacitance and Break Down Voltage are marked on each capacitor As can be seen in this picture Capacitance increases with increasing size Breakdown voltage (6.3V) depends on the material used in construction, usually the electric properties of the material between the plates
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Capacitors in Series and Parallel Capacitors can be found in industry might not be the one what you actually need. Usually manufacturers makes the capacitors with certain capacitance and working voltages. One may always use a combination of capacitors to get what is need. There are two possible combinations in connecting two capacitors 1) Series connection : capacitors are connected one after another For both cases: The connected capacitors acts like one capacitor with a capacitance called equivalent capacitance. 2) Parallel Connection: capacitors legs are connected side by side
Capacitors in Series When two capacitors are connected in series and a potential difference V ab =V applied across the capacitors, Q charge will be charged on both capacitors. Because the plates in between just exchange charges Also the potential across a and b must be the sum of ac and cb Let s call these potentials as V, V 1, and V 2 V V1 V2 Q Q By using C= for C1 and C2 : V and V Q By using C= for Ceq : V V Q C eq V 1 2 C1 Q C 2 Potential differences must satisfy: V V V 1 2 Q Q Q 1 1 1 = = Ceq C1 C2 Ceq C1 C2 Reciprocal of the equivalent capacitance equals to sum of reciprocals of the individual capacitances
Capacitors in Parallel When two capacitors are connected in parallel and a potential difference V ab =V applied across the capacitors, same potential difference exist along each capacitors that will cause charging Q 1 and Q 2 on each capacitors. Net/total charge charged : Q Q1 Q2 Q By using C= for C and C : Q CV and Q V Q By using C= for Ceq : Q V CV 1 2 1 1 2 2 C Total charge Q must satisfy: Q Q Q eq V 1 2 CV CV+ CV C C+ C eq 1 2 eq 1 2 The equivalent capacitance equals to sum of the individual capacitances
Combinations of Capacitors Comparison 1) Series Connection : Q charge is stored on the capacitors net Q not 2Q Where Q C V eq and equivalent capacitance Potential Differences: and 2) Parallel Connection: V potential difference across each capacitor is same V=V 1 =V 2 Stored charges on each capacitor: Q 1 =C 1 V and Q 2 =C 2 V and equivalent capacitance Net/Total charged stored: Q = C eq V= Q 1 + Q 2
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Page 796 In the network some of them is parallel and some of them are in series, we can reduce the network by applying the equivalent capacitance idea until we get one capacitance.
Energy Stored in Capacitors Derivation So far, we have called the capacitors as the electric devices used to store electric energy. Now, lets try to calculate an expression for the energy stored in a capacitor by considering the simplest parallel plate capacitor, and keep in mind that this expression will also be valid for any type of capacitor. Consider the plates are initially uncharged Q=0 and the batter will do some work to move the charges from one plate to other plate until the charge becomes Q=Q. As we know that the V potential difference across a capacitor increases while the Q increases, so the work needs to be done will be increasing to bring extra charge. For this reason, we need to use an integral expression to calculate the work. This integral, will gives us the total work needs to be done to charge the capacitor with a charge Q, also equals to potential energy stored in the capacitor. Usually this is the work done by the battery. Now, lets say at any time, charge on the capacitor is q and a dw work needs to be done to bring extra dq onto q charge. When there is dq charge on capacitor v (pot. diff) = dq/c.
Energy Stored in Capacitors Electric Field Energy The expression for the potential energy for a charged capacitor: where C in Farads, V in Volts, and Q in Coulombs U in joule This expression can be used to calculate the energy stored in any type of capacitor. Energy Stored in Electric Field: Electric Field Energy Energy stored in a capacitor can be used to define the energy stored per unit volume by means of electric field. In order to keep it simple let s consider a parallelplate capacitor charged with charge Q. +Q Q Q/ A A E and C 0 and V Ed d 0 0 Substitute C and V into above equation and after arranging it: Ad is the volume between the plates: u 1 0E 2 2 1 2 U 0E ( Ad) 2 Energy Stored by E. Field per unit volume
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Dielectrics nonconducting materials Space between the conductors inside a the capacitors are mostly filled with dielectric materials. ***This process mainly serves three functions*** 1) Helps to maintain the conductors in a small separation without contact. 2) It increases the breakdown value for the capacitor (max voltage it can hold). 3) Increases the capacitance of capacitor comparing to formed in vacuum. Let s try to understand the 2 nd and 3 rd functions of the dielectric in a capacitor Consider a charged capacitor then insert a dielectric material between the conductors. A voltmeter is connected along the capacitor to keep track of the change in the potential. Initially: Q and V 0 C 0 = Q/V 0 Finally: Q and V where V < V 0 C=Q/V Because V < V 0 then C > C 0. Potential decreases with the dielectric. Capacitance increases with the dielectric.
Dielectrics Dielectric Constant The capacitor filled with dielectric increases, using this idea dielectric constant, K is defined as a measure of this increase comparing to C in the vacuum. C KC0 where C0 is the capacitance in the vacuum. C K called dielectric constant. ( no unit unitless quantity) C 0 Page 801 K=1 for vacuum and something greater than 1 for other dielectrics.
Induced Charge and Polarization Let s now try to explain why the potential difference gets smaller when a dielectric is inserted into a capacitor, which is eventually results an increase in capacitance The experiment has revealed that the potential difference across the plates in a capacitor decreases. If we remember that E = V/d then the electric field is also expected to be decreases. Let s analysis the field between the parallel plates. E E0 K where E is the electric field in the vacuum 0 The electric field gets smaller because of the polarization (charge separation) effect in the dielectric material. Where σ i represents the induced charged density on the dielectric. We can determine this σ i in terms of K dielectric constant. Using Gauss s Law E= σ net /ε 0 Induced charge density: E0 E K solving for along with the above equations for E and i 0 E
Electric Field inside a Dielectric E K 0 inside the dielectric electric field: where which can be written as E K 0 Recall that ε 0 is the permittivity of the vacuum/free space. E E 0 ε =Kε 0 quantity is called as the permittivity of the dielectric. In terms of, the electric field inside... E So, you can use any previous equation by replacing ε 0 with ε 0 or in the energy expression of the electric field
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Dielectric Breakdown When we put a dielectric in an external electric field, it causes polarization. If the electric field gets bigger, the polarization (charge separation) gets bigger. Once the electric field gets enough large then the electrons inside dielectric starts to become ripped and dielectric becomes like conductor (charges start to flow through the dielectric just like it is a conductor). This maximum electric field that a dielectric can sustain/tolerate is called Electric Strength of the dielectric material. For dry air/vacuum ~ 3x10 6 V/m.
Gauss s Law inside a Dielectric When we are studying electric field inside a dielectric and if we need to use Gauss s Law then we can use the Gauss s Law in the following form Where K is the dielectric constant and Q encl free accounts for the free charges, not the induced charges. Then E becomes the electric field inside the dielectric. Or we may just replace ε 0 with ε and use the following formula Recall that ε=k ε 0 E da = Qencl-free EXAMPLE: Electric Field between parallel plates filled with a dielectric, K Considering the closed integral through cylindrical (like a can) Gauss surface A EA E K 0
Page 808 Space between conductors is filled with dielectric material with dielectric constant K E between the plates =?
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RECOMMENDED END OF CHAPTER 24 QUESTIONS AND PROBLEMS 1,3,4,5,6,7,8,9,12,14,15, and 17 1,2,3,4,5,6,8,9,10,11,12,13,14,16,17,18,19,20,21, 22,23,24,25,26,30,31,33,35,36,37,41,43,45, and 46 53,54,55,57,64,65,66,68, and 72