3Simple Capacitor Circuit Direction of arrows is opposite of the direction of electron motionSorry. It’s historical. There is nothing I can do.QCVQ = VCVCThis “Q” really means that the battery has moved charge Q from one plate to the other, so that one plate holds +Q and the other -Q.
4Parallel Capacitor Circuit QtotalC1C2VQ1 = C1VQ2 = C2VKey point: V is the same for both capacitorsKey Point: Qtotal = Q1 + Q2 = VC1 + VC2 = V(C1 + C2)Ctotal = C1 + C2
5Series Capacitor Circuit QQ = VCtotalC1C2V1V2VVQKey point: Q is the same for both capacitorsKey point: Q = VCtotal = V1C1 = V2C2Also: V = V1 + V2Q/Ctotal = Q/C1 + Q/C21CtotalC1C2+=
6Capacitor Summary Series Parallel Wiring Voltage Current Capacitance Every electron that goes through one must go through the otherEach resistor on a different wire.WiringSame for each capacitor.Vtotal = V1 = V2VoltageDifferent for each capacitor.Vtotal = V1 + V2Same for each capacitorItotal = I1 = I2Different for each capacitorItotal = I1 + I2CurrentDecreases1/Ceq = 1/C1 + 1/C2IncreasesCeq = C1 + C2Capacitance
7Example 8.1 (Capacitors in Series) Given the circuit to the left:What is Q1?What is V1?C1 =3uFV =12VC2 =9uFConceptual Idea:Plan:Find equivalent capacitanceUse knowledge that in series Q1 = Q2 = QtotFind V using Q = CV
8Example 8.2 (Capacitors in Parallel) Given the circuit to the left:What is Q1?What is Q2?C2 =9uFC1 =3uFV =12VConceptual Idea:Plan:Find equivalent capacitanceUse knowledge that in parallel V1 = V2Find Q using Q = CV
9CheckPoint: Three Capacitor Configurations The three configurations shown below are constructed using identical capacitors. Which of these configurations has lowest total capacitance?BCCA1/Ctotal = 1/C + 1/C= 2/CCtotal = CCtotal = 2CCtotal = C/2
10CheckPoint: Two Capacitor Configurations The two configurations shown below are constructed using identical capacitors. Which of these configurations has the lowest overall capacitance?BCACCright = C/2Cleft = C/2Ctotal = CCtotal = Cleft + CrightCtotal = CABBoth configurations have the same capacitance
11CheckPoint: Capacitor Network A circuit consists of three unequal capacitors C1, C2, and C3 which are connected to a battery of voltage V0. The capacitance of C2 is twice that of C1. The capacitance of C3 is three times that of C1. The capacitors obtain charges Q1, Q2, and Q3.C1C2C3V0V1Q1V2Q2V3Q3Compare Q1, Q2, and Q3.Q1 > Q3 > Q2Q1 > Q2 > Q3Q1 > Q2 = Q3Q1 = Q2 = Q3Q1 < Q2 = Q3
12Example 8.3 (Capacitor Network) C3 =12uFGiven the circuit to the left:What is Q1?What is Q3?C1 =3uFC2 =11uFV =12CC5 =9uFC4 =6uFConceptual Idea:Find V at each capacitor and then Q .Plan:Break circuit down into elements that are in parrallel or in series.Find equivalent capacitanceWork backwards to find DV across each oneFine Q using Q = CV
14Energy in a Capacitor +Q C V U = 1/2QV = 1/2CV2 -Q = 1/2Q2/C In Prelecture 7 we calculated the work done to move charge Q from one plate to another:C+QVU = 1/2QV= 1/2CV2Since Q = VC= 1/2Q2/C-QThis is potential energy waiting to be used…
15Messing with Capacitors If connected to a battery V stays constant If isolated then total Q stays constantVV1 = VQ1 = C1V1= k CV = k QC1 = k CQ1 = QV1 = Q1/C1= Q/k C = V /kC1 = k C
16DielectricsC1 = k C0VQ1 = VC1C0VQ0 = VC0By adding a dielectric you are just making a new capacitor with larger capacitance (factor of k)
17CheckPoint: Capacitors and Dielectrics 1 Two identical parallel plate capacitors are given the same charge Q, after which they are disconnected from the battery. After C2 has been charged and disconnected, it is filled with a dielectric.Compare the voltages of the two capacitors.V1 > V2V1 = V2V1 < V2
18CheckPoint: Capacitors and Dielectrics 2 Two identical parallel plate capacitors are given the same charge Q, after which they are disconnected from the battery. After C2 has been charged and disconnected, it is filled with a dielectric.Compare the potential energy stored by the two capacitors.U1 > U2U1 = U2U1 < U2
19CheckPoint: Capacitors and Dielectrics 3 The two capacitors are now connected to each other by wires as shown. How will the charge redistribute itself, if at all?The charges will flow so that the charge on C1 will become equal to the charge on C2.The charges will flow so that the energy stored in C1 will become equal to the energy stored in C2The charges will flow so that the potential difference across C1 will become the same as the potential difference across C2.No charges will flow. The charge on the capacitors will remain what it was before they were connected.
20Example 8.4 (Partial Dielectric) VC0x0An air-gap capacitor, having capacitance C0 and width x0 is connected to a battery of voltage V.Vkx0/4A dielectric (k ) of width x0/4 is inserted into the gap as shown.What is Qf, the final charge on the capacitor?Conceptual Analysis:The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.Strategic Analysis:Think of new capacitor as two capacitors in parallelCalculate new capacitance CApply definition of capacitance to determine Q
21Calculation A1 =3/4 Ao A2 =1/4 Ao k k The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.