Slide 1HITTSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Chapter 25 Capacitance-IIIn the last lecture: we calculated the capacitance C of a system of two isolated conductors. We also calculated the capacitance for some simple geometries.In this chapter we will cover the following topics:-Methods of connecting capacitors (in series , in parallel). -Equivalent capacitance. -Energy stored in a capacitor. -Behavior of an insulator (a.k.a. dielectric) when placed in the electric field created in the space between the plates of a capacitor. -Gauss’ law in the presence of dielectrics.(25 - 1)HITTA. the work done by the field is positive and the potential energy of the electron-field system increasesB. the work done by the field is negative and the potential energy of the electron-field system increasesC. the work done by the field is positive and the potential energy of the electron-field system decreasesD. the work done by the field is negative and the potential energy of the electron-field system decreasesE. the work done by the field is positive and the potential energy of the electron-field system does not changeAn electron moves from point i to point f, in the direction of a uniform electric field. During this displacement:ijThis means that if we apply the same voltage across the capacitors in fig.a and fig.b (either right or left) by connectingto a battery, the same charge is provided by the battery. Alternatively,iVqf we place the same charge on plates of the capacitors in fig.a and fig.b (either right or left), the voltage across them is identical. This can be stated in the following manner: If we place theqV capacitorcombination and the equivalent capacitor in separate black boxes, by doing electrical mesurements we cannot distinguish between the two. Consider the combination of capacitorsshown in the figure to the left and to the right (upper part). We will substitutethese combinations of capacitor with a single capacitor eqCEquivalent Capacitorthat is "electrically equivalent" to the capacitor group it substitutes.(25 - 9)The fig.a we show three capacitors connected in parallel. This means that the plate of each capacitor is connectedto the terminals of a battery of voltage . We will substitute VCapacitors in parallelthe parallel combination of fig.a with a single equivalent capacitor shown in fig.b which is also connectedto an identical battery1 1 1 2 2 23 3 3 1 2 3 1The three capacitors have the across their plates.The charge on is: . The charge on is: .The charge on is: . The net charge C q C V C q C VC q C V q q q q C= == = + + =same potential difference V( )( )2 31 2 31 2 31 21The equivalent capacitance For a parallel combination of n capacitors is given by the expression: ...ne neqq jjC C VC CC C C C CC VqC C C CV V==+ ++ += = = + ++ + + =�1 2 3eqC C C C= + +(25 - 10)The fig.a we show three capacitors connected in series. This means that one capacitor is connected after the other.The combination is connected to the terminals of a battery of volCapacitors in seriestage . We will substitute the series combination of fig.a with a single equivalent capacitor shown in fig.b which is also connected to an identical battery.The three capacitors have theVsame charg1 1 12 2 23 3 31 2 3 on their plates.The voltage across is: / . The voltage across is: / . The voltage across is: / .The net voltage across the combina tion Thus we have: C V q CC V q CC V q CV V V VV q==== + +=e q1 2 31 2 31 1 1The equivalent capacitance 1 1 1eqC C Cq qCVqC C C� �+ +� �� �= = �� �+ +� �� �1 2 31 1 1 1eqC C C C= + +(25 - 11)In general a capacitor systemmay consist of smaller capacitorgroups that can be identified asconnected "in parallel" or "in series" More complex capacitor systems12 1 21 212 3In the example of the figure and in fig.a are connected . They can be substituted by the equivalent capacitor as shown in fig.b. Capin parallelacitors and in fig.b are cC C CC CC C= +123123123 12 3onnected .They can be substituted by a single capacitor as shown in fig.c is given by the equation: in series1 1 1 CCC C C= +(25 - 12)-----+++++dq'q'-q'V'q'q V'VChargeVoltageOABConsider a capacitor which is has a charge .We can calculate the work required to chargethe capacitor by assuming that we transfer a charge from the negativeC qWdq�Energy stored an an electric field plate to the positive plate.We assume that the capacitor charge is and thecorresponding voltage . The work required for the charge transfer is given by: We continue this procqV dWqdW V dq dqC����� �= =0ess till the capacitor charge is 1equal to . The total work qq W V dq q dqC�� � �= =� �2 2021 If we substitute we get: or 2Work can also be calculated by determining the area of triangle OAB which is equal to . Area2 2 22qq CVqW q CVCW AVqV dqq WVW WC�� �= = =� �� ���= == =�(25 - 13)2 2The work spent to charge a capacitor is stored in the form of potential energy that can be retrieved when is capacitor is discharged. Thus 2 2WU Wq CV qVUC== = =Potential energy stored in a capacitor 2 We can ask the question: where is the potential energy of a chargedcapacitor stored? The answer is counter intuitive. The energy is stored in the space between the capacitor plateEnergy densitys where a uniform electric field / is generated by the capacitor charges.In other words the electric field can store energy in empty space!E V d=2 22 2q CVUC= =-----+++++q-qdErA A2 2We define as energy densiry (symbol ) the potential energy per unit volume. The volume between the plates is: where is the plate areaThus the energy density 2 2 2oUu uVV V Ad AU CV V VuAd Ad Ad de==�= = = =�22This result, derived for the parallel plate capacitor holds in general 2oEe�=� ��22oEue=(25 - 14)q-qq'qq-q'-q-qVVVV'In 1837 Michael Faraday investigated what happens to thecapacitance of a capacitor when the gap between the plates is completely filled with an insulator (a.k.a. dielectri CCapacitor with a dielectricc)Faraday discovered that the new
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