Lightning Review16.7 The parallel-plate capacitorProblem: parallel-plate capacitor16.8 Combinations of capacitorsa. Parallel combinationParallel combination: notesProblem: parallel combination of capacitorsb. Series combinationSeries combination: notesProblem: series combination of capacitors16.9 Energy stored in a charged capacitorExample: electric field energy in parallel-plate capacitorExample: stored energy16.10 Capacitors with dielectricsCapacitors with dielectrics - notesExample119/15/20039/15/2003General Physics (PHY 2140)Lecture 6Lecture 6¾ Electrostatics9Capacitance and capacitors9 parallel-plate capacitor9 capacitors in electric circuits9 energy stored in a capacitor9 capacitors with dielectricsChapter 16http://www.physics.wayne.edu/~apetrov/PHY2140/229/15/20039/15/2003Lightning ReviewLightning ReviewLast lecture:1.1.Potential and potential energyPotential and potential energy99Potential and potential energy of a system of point chargesPotential and potential energy of a system of point charges99Superposition principle for potential (algebraic sum)Superposition principle for potential (algebraic sum)99Potentials and charged conductors (V is the same in a conductor)Potentials and charged conductors (V is the same in a conductor)99EquipotentialEquipotentialsurfaces (surfaces of constant potential)surfaces (surfaces of constant potential)2.2.Capacitance and capacitorsCapacitance and capacitorsReview Problem:A cylindrical piece of insulating material is placed in an external electric field, as shown. The net electricflux passing through the surface of the cylinder is a. negativeb. positivec. zeroeqVkr=QCV=∆339/15/20039/15/200316.7 The parallel16.7 The parallel--plate capacitorplate capacitorThe capacitance of a device The capacitance of a device depends on the geometric depends on the geometric arrangement of the conductorsarrangement of the conductorswhere where AAis the area of one of is the area of one of the plates, the plates, ddis the separation, is the separation, εε00is a constant (is a constant (permittivity of permittivity of free space)free space),,εε00= 8.85= 8.85××1010--12 12 CC22/N/N··mm2AA+Qd0ACdε=-Q014ekπε=2449/15/20039/15/2003Problem: parallelProblem: parallel--plate capacitorplate capacitorA parallel plate capacitor has plates 2.00 m2in area, separated by a distance of 5.00 mm. A potential difference of 10,000 V is applied across the capacitor. Determinethe capacitancethe charge on each plate559/15/20039/15/2003A parallel plate capacitor has plates 2.00 m2in area, separated by a distance of 5.00 mm. A potential difference of 10,000 V is applied across the capacitor. Determinethe capacitancethe charge on each plateSolution:Given:∆V=10,000 VA = 2.00 m2d = 5.00 mm Find:C=?Q=?Since we are dealing with the parallel-plate capacitor, the capacitance can be found as()212 2 20392.008.85 105.00 103.54 10 3.54AmCCNmdmFnFε−−−==× ⋅×=× =Once the capacitance is known, the charge can be found from the definition of a capacitance via charge and potential difference:()()953.54 10 10000 3.54 10QCV F V C−−=∆= × = ×669/15/20039/15/200316.8 Combinations of capacitors16.8 Combinations of capacitorsIt is very often that more than one capacitor is used in an It is very often that more than one capacitor is used in an electric circuitelectric circuitWe would have to learn how to compute the equivalent We would have to learn how to compute the equivalent capacitance of certain combinations of capacitorscapacitance of certain combinations of capacitorsC1C2C3C5C1C2C3C4779/15/20039/15/2003a. Parallel combinationa. Parallel combinationConnecting a battery to the parallel combination of capacitors is equivalent to introducing the same potential difference for both capacitors, +Q1−Q1C1+Q2−Q2C2V=Vabab12VVV==A total charge transferred to the system from the battery is the sum of charges of the two capacitors,12QQ Q+=1112 2212 1 2 1 212121212eqeqQCVQ CVQQ QQ QQQCVV VVVVQQ QCCCVVV==+≡= = + = ++==+==12eqCCC=+By definition,Thus, Ceqwould be889/15/20039/15/2003Parallel combination: notesParallel combination: notesAnalogous formula is true for any number of capacitors,Analogous formula is true for any number of capacitors,It follows that the equivalent capacitance of a parallel It follows that the equivalent capacitance of a parallel combination of capacitors is greater than any of the combination of capacitors is greater than any of the individual capacitors123...eqCCCC=+++(parallel combination)individual capacitors999/15/20039/15/2003Problem: parallel combination of capacitorsProblem: parallel combination of capacitorsA 3 µF capacitor and a 6 µF capacitor are connected in parallel across an 18 V battery. Determine the equivalent capacitance and total charge deposited.10109/15/20039/15/2003A 3 µF capacitor and a 6 µF capacitor are connected in parallel across an 18 V battery. Determine the equivalent capacitance and total charge deposited.+Q1−Q1C1+Q2−Q2C2V=VababGiven:V = 18 VC1= 3 µFC2= 6 µFFind:Ceq=?Q=?First determine equivalent capacitance of C1 and C2:12 1 29CCC Fµ=+=Next, determine the charge()()64910 18 1.610QCV F V C−−=∆= × = ×11119/15/20039/15/2003b. Series combinationb. Series combinationConnecting a battery to the serial combination of capacitors is equivalent to introducing the same charge for both capacitors, +Q1−Q1C1+Q2−Q2C2V=Vabacb12QQQ==A voltage induced in the system from the battery is the sum of potential differences across the individual capacitors,12VVV=+1112 2212 1 2 1 2121212121111eqeqQCVQCVVV V V V VVCQQ QQQQQQQVV VCCC==+≡= = + = +===++=12111eqCCC=+By definition,Thus, Ceqwould be12129/15/20039/15/2003Series combination: notesSeries combination: notesAnalogous formula is true for any number of capacitors,Analogous formula is true for any number of capacitors,It follows that the equivalent capacitance of a series It follows that the equivalent capacitance of a series combination of capacitors is always less than any of the combination of capacitors is always less than any of the individual capacitance in the combination1231111...eqCCCC=+++(series combination)individual capacitance in the combination13139/15/20039/15/2003Problem: series combination of capacitorsProblem: series combination of capacitorsA 3 µF capacitor and a 6 µF capacitor are connected in series across an 18 V battery. Determine the equivalent capacitance.14149/15/20039/15/2003A 3 µF capacitor and a 6 µF capacitor are connected