Thursday, Feb. 17, 2011 1PHYS 1444-02 Dr. Andrew BrandtPHYS 1444 – Section 02Lecture #8• Chapter 24Thursday Feb 17, 2011Dr. Andrew Brandt• Capacitors and Capacitance• Capacitors in Series or Parallel• Electric Energy StorageHW4 Ch 24 will be assigned tomorrow, due next Thurs 24thHW5 Ch 25 will be due Thurs Mar. 3Mar 3 will be ½ period review***Test 1 will be Tues. Mar. 8 on ch 21-25***Thursday, Feb. 17, 2011 2PHYS 1444-02 Dr. Andrew BrandtExample 24 – 3Spherical capacitor: A spherical capacitor consists of two thin concentric spherical conducting shells, of radius raand rb, as in the figure. The inner shell carries a uniformly distributed charge Q on its surface and the outer shell an equal but opposite charge –Q. Determine the capacitance of this configuration. Using Gauss’ law, the electric field outside a uniformly charged conducting sphere is 204QErSo the potential difference between a and b isbbaaV E dl C Thus capacitance isbaE dr 204baQdrr200144barbarQQdrrr 0114baQrr04abbarrQrrQV04abbaQrrQrr04baabrrrr3PHYS 1444-02 Dr. Andrew BrandtCapacitor Cont’d• A single isolated conductor can be said to have a capacitance, C.• C can still be defined as the ratio of the charge to absolute potential V on the conductor.– So Q=CV.• The potential of a single conducting sphere of radius rbcan be obtained as• So its capacitance is• Although it has capacitance, a single conductor is not considered to be a capacitor, as a second nearby charge is required to form a capacitor.V ar 0114baQrr04bQrwhere0/4bC Q V rThursday, Feb. 17, 2011 4PHYS 1444-02 Dr. Andrew BrandtCapacitors in Series or Parallel• Capacitors are used in may electric circuits• What is an electric circuit?– A closed path of conductors, usually wires connecting capacitors and other electrical devices, in which • charges can flow• there is a voltage source such as a battery• Capacitors can be connected in various ways.– In parallel and in Series or in combination5Capacitors in Parallel• Parallel arrangement provides the same voltage across all the capacitors. – Left hand plates are at Vaand right hand plates are at Vb– So each capacitor plate acquires charges given by the formula• Q1=C1V, Q2=C2V, and Q3=C3V• The total charge Q that must leave battery is then– Q=Q1+Q2+Q3=V(C1+C2+C3)• Consider that the three capacitors behave like an equivalent one– Q=CeqV= V(C1+C2+C3)• Thus the equivalent capacitance in parallel is 1 2 3eqC C C C What is the net effect? The capacitance increases!!!Thursday, Feb. 17, 2011 6PHYS 1444-02 Dr. Andrew BrandtCapacitors in Series• Series arrangement is more “interesting”– When battery is connected, +Q flows to the left plate of C1and –Q flows to the right plate of C3– This induces opposite sign charges on the other plates.– Since the capacitor in the middle is originally neutral, charges get induced to neutralize the induced charges– So the charge on each capacitor is the same value, Q. (Same charge)• Consider that the three capacitors behave like an equivalent one– Q=CeqV V=Q/Ceq• The total voltage V across the three capacitors in series must be equal to the sum of the voltages across each capacitor. – V=V1+V2+V3=(Q/C1+Q/C2+Q/C3)• Putting all these together, we obtain:• V=Q/Ceq=Q(1/C1+1/C2+1/C3)• Thus the equivalent capacitance is 1 2 31 1 1 1eqC C C C What is the net effect? The total capacitance is smaller than the smallest C!!!Thursday, Feb. 17, 2011 7PHYS 1444-02 Dr. Andrew BrandtExample 24 – 4Equivalent Capacitor: Determine the capacitance of a single capacitor that will have the same effect as the combination shown in the figure. Take C1=C2=C3=C.We should do these first!! Now the equivalent capacitor is in series with C1. How? These are in parallel so the equivalent capacitance is: eqC 1eqC23eqCC Solve for Ceq32CC2C1211eqCC112CC32CThursday, Feb. 17, 2011 8PHYS 1444-02 Dr. Andrew BrandtElectric Energy Storage• A charged capacitor stores energy.– The stored energy is the work done to charge it.• The net effect of charging a capacitor is removing one type of charge (+ or -) from a plate and moving it to the other plate.– Battery does this when it is connected to a capacitor. • Capacitors do not charge immediately. – Initially when the capacitor is uncharged, no work is necessary to move the first bit of charge. Why?• Since there is no charge, there is no field that the external work needs to overcome.– When some charge is on each plate, it requires work to add more charge due to electric repulsion.Thursday, Feb. 17, 2011 9PHYS 1444-02 Dr. Andrew BrandtElectric Energy Storage• What work is needed to add a small amount of charge (dq) when the potential difference across the plates is V?• Since V=q/C, the work needed to store total charge Q is • Thus, the energy stored in a capacitor when the capacitor carries charges +Q and –Q is• Since Q=CV, we can rewriteW 22QUCU 22QC212CV 12QV0QVdq 01QqdqC22QCdW=VdqThursday, Feb. 17, 2011 10PHYS 1444-02 Dr. Andrew BrandtExample 24 – 7Energy store in a capacitor: A camera flash unit stores energy in a 150mF capacitor at 200V. How much electric energy can be stored?So we use the one with C and V: Umm.. Which one? Use the formula for stored energy. What do we know from the problem? C and V 212U CV 22611150 10 200 3.022U CV F V J How do we get J from FV2? 2FV 2CVVCV JCCJ11PHYS 1444-02 Dr. Andrew BrandtElectric Energy Density• The energy stored in a capacitor can be considered as being stored in the electric field between the two plates• For a uniform field E between two plates, V=Ed and C=0A/d• Thus the stored energy is• Since Ad is the gap volume, we can obtain the energy density, stored energy per unit volume, as U 212CV 2012AEdd2012E Ad2012uEElectric energy stored per unit volume in any region of space is proportional to the square of the electric field in that region.Valid for plates with a vacuum gapThursday, Feb. 17, 2011
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