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Physics 2112 Unit 8: Capacitors Today’s Concept: Capacitors in a circuits Dielectrics Energy in capacitors Unit 8, Slide 1 Where we are……. Unit 8, Slide 2 Simple Capacitor Circuit Q V Direction of arrows is opposite of the direction of electron motion V C C Sorry. It’s historical. There is nothing I can do. Q = VC Q This “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. Electricity & Magnetism Lecture 8, Slide 3 Parallel Capacitor Circuit Qtotal V C2 C1 Q1 = C1V Q2 = C2V Qtotal Key point: V is the same for both capacitors Key Point: Qtotal = Q1 + Q2 = VC1 + VC2 = V(C1 + C2) Ctotal = C1 + C2 Electricity & Magnetism Lecture 8, Slide 4 Series Capacitor Circuit Q = VCtotal Q C1 V1 V Q V C2 V2 Q Key point: Q is the same for both capacitors Key point: Q = VCtotal = V1C1 = V2C2 Also: V = V1 + V2 Q/Ctotal = Q/C1 + Q/C2 1 1 1 = + C2 Ctotal C1 Electricity & Magnetism Lecture 8, Slide 5 Capacitor Summary Series Parallel C1 C1 Wiring Voltage Current C2 Every electron that goes through one must go through the other Different for each capacitor. Vtotal = V1 + V2 Same for each capacitor Itotal = I1 = I2 Decreases Capacitance 1/C = 1/C + 1/C eq 1 2 C2 Each resistor on a different wire. Same for each capacitor. Vtotal = V1 = V2 Different for each capacitor Itotal = I1 + I2 Increases Ceq = C1 + C2 Electricity & Magnetism Lecture 9, Slide 6 Example 8.1 (Capacitors in Series) Given the circuit to the left: C1 =3uF V =12V C2 =9uF What is Q1? What is V1? Conceptual Idea: CV = Q Plan: • • • Find equivalent capacitance Use knowledge that in series Q1 = Q2 = Qtot Find V using Q = CV Unit 8, Slide 7 Example 8.2 (Capacitors in Parallel) Given the circuit to the left: C2 =9uF V =12V C1 =3uF What is Q1? What is Q2? Conceptual Idea: CV = Q Plan: • • • Find equivalent capacitance Use knowledge that in parallel V1 = V2 Find Q using Q = CV Unit 8, Slide 8 CheckPoint: Three Capacitor Configurations The three configurations shown below are constructed using identical capacitors. Which of these configurations has lowest total capacitance? C B A C C C C C Ctotal = C 1/Ctotal = 1/C + 1/C Ctotal = 2C = 2/C Ctotal = C/2 Electricity & Magnetism Lecture 8, Slide 9 CheckPoint: Two Capacitor Configurations The two configurations shown below are constructed using identical capacitors. Which of these configurations has the lowest overall capacitance? B A C C C C C Cleft = C/2 Ctotal = C A. B. C. Cright = C/2 Ctotal = Cleft + Cright C =C total A B Both configurations have the same capacitance Electricity & Magnetism Lecture 8, Slide 10 CheckPoint: 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. Q2 C2 Compare Q1, Q2, and Q3. A. Q1 > Q3 > Q2 V0 B. Q1 > Q2 > Q3 C. Q1 > Q2 = Q3 D. Q1 = Q2 = Q3 E. Q1 < Q2 = Q3 V2 C1 V1 Q1 V3 C3 Q3 Unit 8, Slide 11 Example 8.3 (Capacitor Network) C3 =12uF Given the circuit to the left: C1 =3uF C2 =11uF What is Q1? V =12C What is Q3? C5 =9uF C4 =6uF Conceptual Idea: CV = Q Find V at each capacitor and then Q . Plan: • • • • Break circuit down into elements that are in parrallel or in series. Find equivalent capacitance Work backwards to find DV across each one Fine Q using Q = CV Unit 8, Slide 12 Example 8.1 (Capacitor Network) C3 C4 C3 C5 C1234 C2 C1 C2 C1 C34 C5 C12345 C5 Unit 8, Slide 13 Energy in a Capacitor In Prelecture 7 we calculated the work done to move charge Q from one plate to another: C +Q V -Q U = 1/2QV = 1/2CV2 = 1/2Q2/C Since Q = VC This is potential energy waiting to be used… Electricity & Magnetism Lecture 8, Slide 14 Messing with Capacitors If connected to a battery V stays constant V1 = V V C1 = k C Q1 = C1V1 = k CV = k Q If isolated then total Q stays constant Q1 = Q C1 = k C V1 = Q1/C1 = Q/k C = V /k Electricity & Magnetism Lecture 8, Slide 15 Dielectrics C1 = k C0 C0 V Q0 = VC0 Q1 = VC1 V By adding a dielectric you are just making a new capacitor with larger capacitance (factor of k) Electricity & Magnetism Lecture 8, Slide 16 CheckPoint: 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. A. V1 > V2 B. V1 = V2 C. V1 < V2 Electricity & Magnetism Lecture 8, Slide 17 CheckPoint: 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. A. U1 > U2 B. U1 = U2 C. U1 < U2 Electricity & Magnetism Lecture 8, Slide 18 CheckPoint: 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? A. The charges will flow so that the charge on C1 will become equal to the charge on C2. B. The charges will flow so that the energy stored in C1 will become equal to the energy stored in C2 C. The charges will flow so that the potential difference across C1 will become the same as the potential difference across C2. D. No charges will flow. The charge on the capacitors will remain what it was before they were connected. Electricity & Magnetism Lecture 8, Slide 19 Example 8.4 (Partial Dielectric) C0 V x0 V An air-gap capacitor, having capacitance k C0 and width x0 is connected to a x0/4 battery of voltage V. A dielectric (k ) of width x0/4 is inserted into the gap as shown. What is Qf, the final charge on the capacitor? Conceptual Analysis: C Q V Strategic Analysis: Think of new capacitor as two capacitors in parallel Calculate new capacitance C Apply definition of capacitance to determine Q Electricity & Magnetism Lecture 8, Slide 20 Calculation k k A1 =3/4 Ao A2 =1/4 Ao Electricity & Magnetism Lecture 8, Slide 21