1February 2, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 14February 2, 2005 Physics for Scientists&Engineers 2 2ReviewReview The capacitance of a spherical capacitor is•r1 is the radius of the inner sphere•r2 is the radius of the outer sphere The capacitance of an isolated spherical conductor is•R is the radius of the sphereC = 4!"0r1r2r2# r1C = 4!"0RFebruary 2, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2) The equivalent capacitance for n capacitors inparallel is The equivalent capacitance for n capacitors inseries isCeq= Cii =1n!1Ceq=1Cii =1n!February 2, 2005 Physics for Scientists&Engineers 2 4 A battery must do work to charge a capacitor We can think of this work as changing the electric potential energy ofthe capacitor The differential work dW done by a battery with voltage V to put adifferential charge dq on a capacitor with capacitance C is The total work required to bring the capacitor to its full charge q is This work is stored as electric potential energyEnergy Stored in CapacitorsEnergy Stored in CapacitorsdW = Vdq =qCdqWt= dW!=qCdq0qt!=12qt2CU =12q2C=12CV2=12qV2February 2, 2005 Physics for Scientists&Engineers 2 5 We can define the energy density, u, as the electric potential energyper unit volume Taking the special case of a parallel plate capacitor that has no fringefield, the volume between the plates is the area of each plate times thedistance between the plates, Ad Inserting our formula for the capacitance of a parallel plate capacitorwe getEnergyEnergy Density in CapacitorsDensity in Capacitorsu =Uvolumeu =UAd=12CV2Ad=CV22Adu =!0Ad"#$%&'V22Ad=12!0Vd"#$%&'2February 2, 2005 Physics for Scientists&Engineers 2 6 Recognizing that V/d is the magnitude of the electric field,E, we obtain an expression for the electric potential energydensity for parallel plate capacitor This result, which is specific to the parallel plate capacitor,is in fact much more general This equation holds for all electric fields produced in anyway• Can be used to describe the electric potential energy stored in anelectric field per unit volume occupied by that fieldEnergyEnergy Density in Capacitors (2)Density in Capacitors (2)u =12!0E2February 2, 2005 Physics for Scientists&Engineers 2 7Example: ThundercloudExample: Thundercloud Suppose a thundercloud with a width of 2.0 kmand a length of 3.0 km hovers over a flat area,at an altitude of 500 m and carries a chargeof 160 C.Question 1:• What is the potential difference betweenthe cloud and the ground?Question 2:• Knowing that lightning strikes requireelectric field strengths of approximately2.5 MV/m, are these conditions sufficientfor a lightning strike?Question 3:• What is the total electrical energy contained in this cloud?February 2, 2005 Physics for Scientists&Engineers 2 8Example: Thundercloud (2)Example: Thundercloud (2)Question 1 We can approximate the cloud-ground system as a parallel platecapacitor whose capacitance is The charge carried by the cloud is 160 C, which means that each “plate”of the capacitor has a charge of 80 C 720 million volts!C =!0Ad=(8.85·10-12 F/m)(2000 m)(3000 m)500 m= 0.11 µFV =qC=80 C0.11 µF= 7.2 !108 V3February 2, 2005 Physics for Scientists&Engineers 2 9Example: Thundercloud (3)Example: Thundercloud (3)Question 2 We know the potential difference between the cloud and ground so wecan calculate the electric field Which is lower than 2.5 MV/m, so no lightning cloud to ground• May have lightning to radio tower or tree….Question 3 The total energy stored in a parallel plate capacitor is• Enough energy to run a 1500 W hair dryer for more than 5000 hoursE =Vd=7.2 !108 V500 m= 1.5 MV/mU =12qV = 0.5(80 C)(7.2 !108 V) = 2.9 ! 1010 JFebruary 2, 2005 Physics for Scientists&Engineers 2 10Capacitors with DielectricsCapacitors with Dielectrics We have been discussing capacitors with air or vacuumbetween the plates However, most real-life capacitors have an insulatingmaterial, called a dielectric, between the two plates The dielectric serves several purposes:• Provides a convenient way to maintain mechanical separationbetween the plates• Provides electrical insulation between the plates• Allows the capacitor to hold a higher voltage• Increases the capacitance of the capacitor• Takes advantage of the molecular structure of the dielectric materialFebruary 2, 2005 Physics for Scientists&Engineers 2 11Capacitors with Dielectrics (2)Capacitors with Dielectrics (2) Placing a dielectric between the plates of a capacitorincreases the capacitance of the capacitor by a numericalfactor called the dielectric constant, κ We can express the capacitance of a capacitor with adielectric with dielectric constant κ between the plates as Where Cair is the capacitance of the capacitor without thedielectric Placing the dielectric between the plates of the capacitorhas the effect of lowering the electric field between theplates and allowing more charge to be stored in thecapacitor.C =!CairFebruary 2, 2005 Physics for Scientists&Engineers 2 12Parallel Plate Capacitor with DielectricParallel Plate Capacitor with Dielectric Placing a dielectric between the plates of a parallelplate capacitor modifies the electric field as The constant ε0 is the electric permittivity of freespace The constant ε is the electric permittivity of thedielectric materialE =Eair!=q!"0A=q"A!="!04February 2, 2005 Physics for Scientists&Engineers 2 13Dielectric StrengthDielectric Strength The dielectric strength of a material measures theability of that material to withstand voltagedifferences If the voltage across a dielectric exceeds thebreakdown potential, the dielectric will break downand begin to conduct charge between the plates Real-life dielectrics enable a capacitor provide agiven capacitance and hold the required voltagewithout breaking down Capacitors are usually specified in terms of theircapacitance and rated voltageFebruary 2, 2005 Physics for Scientists&Engineers 2 14Dielectric ConstantDielectric Constant The dielectric constant of vacuum is defined to be one The dielectric constant of is close to one and we will usethe dielectric constant of air as one in our problems The dielectric constants and dielectric strengths ofcommon materials areMaterial Dielectric Constant Dielectric
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