Physics 2112 Unit 8: Capacitors

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

Where we are…….

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

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

Series Capacitor Circuit Q Q=VCtotal Q Q 1 1 1 = + Ctotal C1 C2 C1 V1 V V C2 V2 Key point: Qis the same for both capacitors Key point: Q =VCtotal=V1C1=V2C2 Also: V=V1+ V2 Q/Ctotal=Q/C1+Q/C2

Capacitor Summary Series Parallel C1 C1 C2 C2 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 Increases Ceq=C1+C2 Decreases 1/Ceq=1/C1+1/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 Vusing 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 Qusing Q = CV

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 1/Ctotal= 1/C+ 1/C Ctotal= 2C Ctotal=C = 2/C Ctotal=C/2

CheckPoint: Two Capacitor Configurations Cleft=C/2 Cright=C/2 Ctotal= C 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 Ctotal= C Ctotal= Cleft+ Cright 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. Q2 C2 • Compare Q1, Q2, and Q3. • Q1 > Q3 > Q2 • Q1 > Q2 > Q3 • Q1 > Q2 = Q3 • Q1 = Q2 = Q3 • Q1 < Q2 = Q3 V0 V2 C3 C1 V1 V3 Q3 Q1

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

Example 8.1 (Capacitor Network) C3 C2 C1 C2 C1 C34 C5 C4 C5 C3 C12345 C1234 C5

Energy in a Capacitor =1/2CV2 SinceQ=VC =1/2Q2/C In Prelecture 7 we calculated the work done to move charge Q from one plate to another: +Q C V U=1/2QV -Q This is potential energy waiting to be used…

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

Dielectrics C1=kC0 C0 V V Q1 =VC1 Q0 =VC0 By adding a dielectric you are just making a new capacitor with larger capacitance (factor ofk)

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.

An air-gap capacitor, having capacitanceC0and width x0 is connected to a battery of voltageV. Example 8.4 (Partial Dielectric) C0 V x0 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: • Strategic Analysis: • Think of new capacitor as two capacitors in parallel • Calculate new capacitance C • Apply definition of capacitance to determine Q

Calculation k k A2 =1/4 Ao A1 =3/4 Ao

Physics 2112 Unit 8: Capacitors