PHY PHY 184 184 Spring 2007 Lecture 13 Title Capacitors 1 30 07 184 Lecture 13 1 Notes Homework 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 4 1 30 07 184 Lecture 13 2 Review The capacitance of a spherical capacitor is r1r2 C 4 0 r2 r1 r1 is the radius of the inner sphere r2 is the radius of the outer sphere The capacitance of an isolated spherical conductor is C 4 0 R R is the radius of the sphere 1 30 07 184 Lecture 13 3 Review 2 The equivalent capacitance for n capacitors in parallel is n Ceq Ci i 1 The equivalent capacitance for n capacitors in series is n 1 1 Ceq i 1 Ci 1 30 07 184 Lecture 13 4 Example System of Capacitors Let s analyze a system of five capacitors If each capacitor has a capacitance of 5 nF what is the capacitance of this system of capacitors 1 30 07 184 Lecture 13 5 System of Capacitors 2 We can see that C1 and C2 are in parallel and that C3 is also in parallel with C1 and C2 We can define C123 C1 C2 C3 15 nF and make a new drawing 1 30 07 184 Lecture 13 6 System of Capacitors 3 We can see that C4 and C123 are in series We can define 1 C1234 1 1 C123C4 C1234 C123 C4 C123 C4 3 75 nF and make a new drawing 1 30 07 184 Lecture 13 7 System of Capacitors 4 We can see that C5 and C1234 are in parallel We can define C12345 C1234 C5 C123C4 C C2 C3 C4 C5 1 C5 8 75 nF C123 C4 C1 C2 C3 C4 And make a new drawing 1 30 07 184 Lecture 13 8 System of Capacitors 5 So the equivalent capacitance of our system of capacitors C12345 5 5 5 5 5 nF 8 75 nF 5 5 5 5 More 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 1 30 07 184 Lecture 13 9 Clicker Question Find the equivalent capacitance Ceq A B C 1 30 07 184 Lecture 13 10 Clicker Question Find the equivalent capacitance Ceq C First Step C1 and C2 are in series Second Step C12 and C3 are in parallel 1 30 07 184 Lecture 13 11 A capacitor stores energy Field Theory The energy belongs to the electric fie 1 30 07 184 Lecture 13 12 Energy Stored in Capacitors A 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 is q dW Vdq dq C The total work required to bring the capacitor to its full charge q is qt q 1 qt2 Wt dW dq 0 C 2C This work is stored as electric potential energy 1 q2 1 1 2 U CV qV 2C 2 2 1 30 07 184 Lecture 13 13 Energy Density in Capacitors We define the energy density u as the electric potential energy per unit volume U u volume Taking 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 Ad 2 2 1 U CV CV u 2 Ad Ad 2Ad Inserting our formula for the capacitance of a parallel plate capacitor we find 0 A 2 2 V 1 V d u 0 2Ad 2 d 1 30 07 184 Lecture 13 14 Energy Density in Capacitors 2 Recognizing that V d is the magnitude of the electric field E we obtain an expression for the electric potential energy density for parallel plate capacitor 1 u 0E2 2 This 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 1 30 07 184 Lecture 13 15 Example isolated conducting sphere An 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 4 0R see previous lecture The energy U stored in a capacitor depends on the charge and the capacitance according to and substituting C 4 0R gives 1 30 07 184 Lecture 13 16 Example isolated conducting sphere An 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 1 u 0E2 2 so we must first find the E field at the surface of the sphere Recall Why 1 30 07 184 Lecture 13 17 Example Thundercloud Suppose a thundercloud with horizontal dimensions of 2 0 km by 3 0 km hovers over a flat area at an altitude of 500 m and carries a charge of 160 C Question 1 What is the potential difference between the cloud and the ground Question 2 Knowing that lightning strikes require electric field strengths of approximately 2 5 MV m are these conditions sufficient for a lightning strike Question 3 What is the total electrical energy contained in this cloud 1 30 07 184 Lecture 13 18 Example Thundercloud 2 Question 1 We can approximate the cloud ground system as a parallel plate capacitor whose capacitance is 0 A 8 85 10 12 F m 2000 m 3000 m C 0 11 F d 500 m The charge carried by the cloud is 160 C which means that the plate surface facing the earth has a charge of 80 C V q 80 C 7 2 10 8 V C 0 11 F 720 million volts 1 30 07 184 Lecture 13 19 Example Thundercloud 3 Question 2 We know the potential difference between the cloud and ground so we can calculate the electric field V 7 2 10 8 V E 1 5 MV m d 500 m E 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 place capacitor is 1 U qV 0 5 80 C 7 2 10 8 V 2 9 1010 J 2 Enough energy to run a 1500 W hair dryer for more than …
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