PHY 184NotesReviewReview (2)Example - System of CapacitorsSystem of Capacitors (2)System of Capacitors (3)System of Capacitors (4)System of Capacitors (5)Clicker QuestionSlide 11Slide 12Energy Stored in CapacitorsEnergy Density in CapacitorsEnergy Density in Capacitors (2)Example - isolated conducting sphereSlide 17Example: ThundercloudExample: Thundercloud (2)Example: Thundercloud (3)Slide 21Slide 22Capacitors with DielectricsCapacitors with Dielectrics (2)Parallel Plate Capacitor with Dielectric1/30/07 184 Lecture 13 1PHY 184PHY 184PHY 184PHY 184Spring 2007Lecture 13Title: Capacitors1/30/07 184 Lecture 13 2NotesNotesNotesNotesHomework Set 3 is done!Homework Set 4 is open and Set 5 opens Thursday morning Midterm 1 will take place in class Thursday, February 8.One 8.5 x 11 inch equation sheet (front and back) is allowed.The exam will cover Chapters 16 - 19 Homework Sets 1 - 41/30/07 184 Lecture 13 3ReviewReviewReviewReviewThe capacitance of a spherical capacitor is•r1 is the radius of the inner sphere•r2 is the radius of the outer sphereThe capacitance of an isolated spherical conductor is•R is the radius of the sphereC 40r1r2r2 r1C 40r1r2r2 r1C 40RC 40R1/30/07 184 Lecture 13 4Review (2)Review (2)Review (2)Review (2)The equivalent capacitance for n capacitors in parallel isThe equivalent capacitance for n capacitors in series isCeq Cii 1nCeq Cii 1n1Ceq1Cii 1n1Ceq1Cii 1n1/30/07 184 Lecture 13 5Example - System of CapacitorsExample - System of CapacitorsExample - System of CapacitorsExample - System of CapacitorsLet’s analyze a system of five capacitorsIf each capacitor has a capacitance of 5 nF, what is the capacitance of this system of capacitors?1/30/07 184 Lecture 13 6System of Capacitors (2)System of Capacitors (2)System of Capacitors (2)System of Capacitors (2)We can see that C1 and C2 are in parallel and that C3 is also in parallel with C1 and C2We can define C123 = C1 + C2 + C3 = 15 nF… and make a new drawing1/30/07 184 Lecture 13 7System of Capacitors (3)System of Capacitors (3)System of Capacitors (3)System of Capacitors (3)We can see that C4 and C123 are in seriesWe can define… and make a new drawing1C12341C1231C4 C1234C123C4C123 C4= 3.75 nF1/30/07 184 Lecture 13 8System of Capacitors (4)System of Capacitors (4)System of Capacitors (4)System of Capacitors (4)We can see that C5 and C1234 are in parallelWe can defineAnd make a new drawingC12345C1234 C5C123C4C123 C4 C5(C1 C2 C3)C4C1 C2 C3 C4 C5= 8.75 nF1/30/07 184 Lecture 13 9System of Capacitors (5)System of Capacitors (5)System of Capacitors (5)System of Capacitors (5)So the equivalent capacitance of our system of capacitorsMore than one half of the total capacitance of this arrangement is provided by C5 alone.This result makes it clear that one has to be careful how one arranges capacitors in circuits.C12345(5 5 5)55 5 5 5 5 nF 8.75 nF1/30/07 184 Lecture 13 10Clicker QuestionClicker QuestionClicker QuestionClicker QuestionFind the equivalent capacitance Ceq A) B) C)1/30/07 184 Lecture 13 11Clicker QuestionClicker QuestionClicker QuestionClicker QuestionFind the equivalent capacitance Ceq C) First Step: C1 and C2 are in seriesSecond Step: C12 and C3 are in parallel1/30/07 184 Lecture 13 12A capacitor stores energy.Field Theory: The energy belongs to the electric field.1/30/07 184 Lecture 13 13A battery must do work to charge a capacitor.We can think of this work as changing the electric potential energy of the capacitor.The differential work dW done by a battery with voltage V to put a differential charge dq on a capacitor with capacitance C isThe total work required to bring the capacitor to its full charge q isThis work is stored as electric potential energyEnergy Stored in CapacitorsEnergy Stored in CapacitorsEnergy Stored in CapacitorsEnergy Stored in CapacitorsdW Vdq qCdqWt dWqCdq0qt12qt2CU 12q2C12CV212qV1/30/07 184 Lecture 13 14We define the energy density, u, as the electric potential energy per unit volumeTaking the ideal case of a parallel plate capacitor that has no fringe field, the volume between the plates is the area of each plate times the distance between the plates, AdInserting our formula for the capacitance of a parallel plate capacitor we findEnergy Density in CapacitorsEnergy Density in CapacitorsEnergy Density in CapacitorsEnergy Density in Capacitorsu Uvolumeu UAd12CV2AdCV22Adu 0AdV22Ad120Vd21/30/07 184 Lecture 13 15Recognizing that V/d is the magnitude of the electric field, E, we obtain an expression for the electric potential energy density for parallel plate capacitorThis result, which we derived for the parallel plate capacitor, is in fact completely general.This equation holds for all electric fields produced in any way•The formula gives the quantity of electric field energy per unit volume.Energy Density in Capacitors (2)Energy Density in Capacitors (2)Energy Density in Capacitors (2)Energy Density in Capacitors (2)u 120E2u 120E21/30/07 184 Lecture 13 16An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC. a) How much potential energy is stored in the electric field of the charged conductor? Key Idea: An isolated sphere has a capacitance of C=40R (see previous lecture). The energy U stored in a capacitor depends on the charge and the capacitance according to Example - isolated conducting sphereExample - isolated conducting sphereExample - isolated conducting sphereExample - isolated conducting sphere… and substituting C=40R gives1/30/07 184 Lecture 13 17An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC. b) What is the field energy density at the surface of the sphere? Key Idea: The energy density u depends on the magnitude of the electric field E according to so we must first find the E field at the surface of the sphere. Recall: Example - isolated conducting sphereExample - isolated conducting sphereExample - isolated conducting sphereExample - isolated conducting sphereu 120E2u 120E2(Why?)1/30/07 184 Lecture 13 18Example: ThundercloudExample:
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