Lecture 4 Capacitance and CapacitorsCapacitanceThe Parallel-Plate CapacitorCapacitorsCombinations of CapacitorsParallel CombinationSlide 7Series CombinationSlide 9Energy Stored in a CapacitorCapacitors with DielectricsDielectric StrengthSummaryLecture 4Capacitance and CapacitorsChapter 16.6  16.10Outline•Definition of Capacitance•Simple Capacitors•Combinations of Capacitors•Capacitors with DielectricsCapacitanceIntroductionThe capacitance C of a capacitor is the ratio of the charge (Q) on either conductor plate to the potential difference (V) between the plates. QC   VGeneral definitionUnits of capacitance are farads (F)1F  1C/1VC(Earth) ~ 1 FC(adult) ~ 150 pF = 150 1012 FHistoryThe Parallel-Plate CapacitorThe capacitance of a parallel-plate capacitor whose plates are separated by air is: AC = є0  dA is the area of one of the platesd is the distance between the platesє0 is permittivity of free spaceMore about capacitorsCapacitorsProblem: A parallel-plane capacitor has an area of A=5cm2 and a plate separation of d=5mm. Find its capacitance. Unit conversion: A = 5 cm2 = 5 104 m2 d = 5 mm = 5 103 mC = є0 A/d = 8.85 1012 C2 / (N m2) 5 104 m2 / 5 103 m = 8.85 1013 C2/(N m)= 8.85 1013 F = 0.885 pFN/C = V/m  C/N = m/V, F=C/VC2/(N m)=C (C/N)/m = C (m/V)/m = C/V = FCombinations of CapacitorsIn real electric circuits capacitors can be connected in various ways.In order to design a circuit with desired capacitance, equivalent capacitance of certain combinations of capacitors can be calculated.There are 2 typical combinations of capacitors:•Parallel combination•Series combinationParallel CombinationParallel Combination•The left plate of each capacitor is connected to the positive terminal of a battery by a wire • the left plates are at the same potential • the potential differences across the capacitors are the same, equal to the voltage of the battery (V).•The charge flow ceases when the voltage across the capacitors equals to that of the battery and the capacitors reach their maximum charge.Q = Q1 + Q2Q1 = C1 VQ2 = C2 VQ = Ceq VCeq V = C1 V + C2 VCeq = C1 + C2ExamplesSeries CombinationSeries CombinationThe magnitude of the charge is the same on all the plates.The equivalent capacitor must have a charge –Q on the right plate and +Q on the left plate. QV =  CeqV = V1 + V2V1 = Q/C1V2 = Q/C2Q Q Q =  + Ceq C1 C21 1 1 =  + Ceq C1 C2ExamplesEnergy Stored in a CapacitorThe work required to move a charge Q through a potential difference V is W = V Q. V = Q/C, Q is the total charge on the capacitor.The voltage on the capacitor linearly increases with the magnitude of the charge.Additional work increases the energy stored.W = ½ Q V = ½ (C V) V = ½C (V)2 = Q2/2CCapacitors with DielectricsA dielectric is an insulating material.The dielectric filling the space between the plates completely increases the capacitance by the factor  > 1, called the dielectric constant.If V0 is the potential difference (voltage) across a capacitor of a capacitance C0 and a charge Q0 in the absence of a dielectric. Filling the capacitor with a dielectric reduces the voltage by the factor  to V, so that V = V0/.C = Q0/V = Q0/V0/ =  Q0/V0 =  C0Dielectric StrengthFor a parallel-plate capacitor: C =  є0 A/d The formula shows that the capacitance can be made very large by decreasing the plate separation.In practice, the lowest value of d is limited by the electric discharge through the dielectric.The discharge occurs when the electric field in the dielectric material reaches its maximum, called dielectric strength.Dielectric strength of air is 3 106 V/m.Summary•Capacitance is defined as the charge over the potential difference•Capacitance of parallel-plate capacitor is directly proportional to the plate area and inversely proportional to the plate separation•The equivalent capacitance of a parallel combination of capacitors equals to the sum of individual capacitances•The inverse equivalent capacitance of a series combination of capacitors equals to the sum of the inverse individual capacitances•Placing a dielectric between the plates of a capacitor increases the capacitance by a factor , called the dielectric