Physics 2112 Unit 8: Capacitors Today’s Concept: Capacitors in a circuits Dielectrics Energy in capacitors

Where we are…….

Simple Capacitor Circuit Direction of arrows is opposite of the direction of electron motion Sorry. It’s historical. There is nothing I can do. Q C V Q = VC V C 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.

Parallel Capacitor Circuit Qtotal C1 C2 V Q1 = C1V Q2 = C2V Key point: V is the same for both capacitors Key Point: Qtotal = Q1 + Q2 = VC1 + VC2 = V(C1 + C2) Ctotal = C1 + C2

Series Capacitor Circuit Q Q = VCtotal C1 C2 V1 V2 V V 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 Ctotal C1 C2 + =

Capacitor Summary Series Parallel Wiring Voltage Current Capacitance Every electron that goes through one must go through the other Each resistor on a different wire. Wiring Same for each capacitor. Vtotal = V1 = V2 Voltage Different for each capacitor. Vtotal = V1 + V2 Same for each capacitor Itotal = I1 = I2 Different for each capacitor Itotal = I1 + I2 Current Decreases 1/Ceq = 1/C1 + 1/C2 Increases Ceq = C1 + C2 Capacitance

Example 8.1 (Capacitors in Series) Given the circuit to the left: What is Q1? What is V1? C1 =3uF V =12V C2 =9uF Conceptual Idea: Plan: Find equivalent capacitance Use knowledge that in series Q1 = Q2 = Qtot Find V using Q = CV

Example 8.2 (Capacitors in Parallel) Given the circuit to the left: What is Q1? What is Q2? C2 =9uF C1 =3uF V =12V Conceptual Idea: Plan: Find equivalent capacitance Use knowledge that in parallel V1 = V2 Find Q using Q = CV

CheckPoint: Three Capacitor Configurations The three configurations shown below are constructed using identical capacitors. Which of these configurations has lowest total capacitance? B C C A 1/Ctotal = 1/C + 1/C = 2/C Ctotal = C Ctotal = 2C Ctotal = C/2

CheckPoint: Two Capacitor Configurations The two configurations shown below are constructed using identical capacitors. Which of these configurations has the lowest overall capacitance? B C A C Cright = C/2 Cleft = C/2 Ctotal = C Ctotal = Cleft + Cright Ctotal = C A B Both configurations have the same capacitance

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. C1 C2 C3 V0 V1 Q1 V2 Q2 V3 Q3 Compare Q1, Q2, and Q3. Q1 > Q3 > Q2 Q1 > Q2 > Q3 Q1 > Q2 = Q3 Q1 = Q2 = Q3 Q1 < Q2 = Q3

Example 8.3 (Capacitor Network) C3 =12uF Given the circuit to the left: What is Q1? What is Q3? C1 =3uF C2 =11uF V =12C C5 =9uF C4 =6uF Conceptual Idea: 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

Example 8.1 (Capacitor Network)

Energy 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 +Q V U = 1/2QV = 1/2CV2 Since Q = VC = 1/2Q2/C -Q This is potential energy waiting to be used…

Messing with Capacitors If connected to a battery V stays constant If isolated then total Q stays constant V V1 = V Q1 = C1V1 = k CV = k Q C1 = k C Q1 = Q V1 = Q1/C1 = Q/k C = V /k C1 = k C

Dielectrics C1 = k C0 V Q1 = VC1 C0 V Q0 = VC0 By adding a dielectric you are just making a new capacitor with larger capacitance (factor of k)

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. V1 > V2 V1 = V2 V1 < V2

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. U1 > U2 U1 = U2 U1 < U2

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? 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 C2 The 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.

Example 8.4 (Partial Dielectric) V C0 x0 An air-gap capacitor, having capacitance C0 and width x0 is connected to a battery of voltage V. V k x0/4 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: 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 parallel Calculate new capacitance C Apply definition of capacitance to determine Q

Calculation 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.