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Lesson 4 Capacitance and Dielectrics Capacitance Capacitors in combination #Series #Parallel Energy stored in the electric field of capacitors and energy density Dielectrics Dielectric Strength Field above surface of charged conductor Field Above Conductor Q s E = Ae = e 0 0 Does not depend on thickness of conductor conductor in electrostatic equilibrium sA = e0 =E E dA = closed cylinder EdA Area A E A dA A sA s \E = = Ae 0 e 0 charge = s E Charged Plates d +Q + -Q - W = Fd =QEd DU = -QEd = U- - U+ \DV = -Ed =V- - V+ PD between Plates Potential drops Ed in going from + to V- is Ed lower than V+ Work Done in Moving Charge How does one make such a separation of charge? Must move positive charge Work is done on positive charge in producing separation +Q F Q -Q Electric Field What forms when we have separation of charge? An Electric Field +Q E -Q The work done on separating charges to fixed positions is stored as potential energy in this electric field, which can thus DO work This arrangement is called a Capacitorb CAPACITOR Moving Charge How do we move charge? With an electric field along a conduction path Picture Charge Separation The charge separation is maintained by removing the conduction path once a charge separation has been produced An electric component that does this is called A Capacitor Capacitor Symbol Battery Symbol + - Can charge a capacitor by Charging itCapacitor connecting to a battery + + - - Plates are conductors Capacitance Equipotential surfaces Let V = P.D. (potential difference) between plates Q (charge on plates) ~ V (why?) Thus Q = CV C is a constant called CAPACITANCE Q C C = = V V SI Units Coulombs = Farads Volts Calculation of Capacitance assume charge Q on plates calculate E between plates using Gauss’ Law From E calculate V Then use C = Q/V Capacitors Electric Field above Plates s Q E= = e0 Ae0 is to plates Q = EAe0 going from positive to negative plate Calculating Capacitance in General DV = V V = - E ds 0 f f i In order that i E d s 0 choose path from + plate to - plate D V = - V ( PD across plates ) Thus V = - Eds (choose path + e0 EA C = - + Eds || to electric field ) for Parallel Plates Capacitor Q EAe0 EA e0 Ae0 = = C= = V Ed d Eds - + for Cylindrical Capacitor Q 2pe0 L = C = b V ln a •a = radius of inner cylinder •b = radius of outer cylinder •L = length of cylinder Combination of Capacitors Combinations of Capacitors in Parallel equilibrium Parallel same electric potential felt by each element Series electric potential felt by the combination is the sum of the potentials across each element Picture Q Q 1 V =Calculation = 2 of Effective C1 Capacitance C2 Total charge = Q = Q1 + Q2 = VC1 + VC2 = VCeq \ Ceq = C1 + C2 In general Ceq = i Ci Combination of Capacitors Series Net charge zero Net charge zero Picture Why are the charges on the plates of equal magnitude ? If net charge inside these Gaussian surfaces is not zero Field lines pass through the surfaces and cause charge to flow Then we do have not equilibrium Calculation of Effective Capacitance I Q Q V total = V1 + Vof = + Calculation 2 Effective C1 C2 Capacitance II 1 1 1 = Q + = Q C2 Ceq C1 In general 1 1 = Ceq Ci i Is this parallel or series? = Question I Is this parallel or series? + - Question II + - Work done in charging capacitor + - in Charging WorkI Done q Capacitor + - q Calculation V q = C if dq of charge is then transfered the work done is dW = V q dq Thus total work done on charging is Q 1 W = V q dq = C 0 Q Q2 1 qdq = = CV 2 2C 2 0 This work is stored as P. E. Energy Density U EnergyDensity= Volume U for parallel plate capacitor= Ad 2 2 CV 1 V 1 = = e 0 = e 0 E 2 2 Ad 2 d 2 Dielectrics Picture Picture Picture Polarization Polarization Induced Electric Field Dielectric Constant Charge Q stays the same, Total electric Field is less, thus P. D. V effective across plates is less Q Q \C = C = V V effective C = C Dielectric Constant 1. 00 Permitivity C= e0 A d A \ C = C = e 0 d A C = e d where, PERMITTIVITY of the dielectric e = e 0 Permitivity in Dielectrics For conductors (not dielectrics ) =e = For regions containing dielectrics all electrostatic equations containing e 0 are replaced by e e . g . Gauss ' Law F= E dA = surface Q e Dielectric Strength The Dielectric Strength of a non conducting material is the value of the Electric Field that causes it to be a conductor. When dielectric strength of air is surpassed we get lightning