Physics 2102 Jonathan Dowling Physics 2102 Lecture 12 MON FEB Capacitance II 25 4 5 Capacitors in Parallel V Constant An ISOLATED wire is an equipotential surface V Constant Capacitors in parallel have SAME potential difference but NOT ALWAYS same charge VAB VCD V Ceq C1 C2 C C1 VA Q2 VB C2 VC VD B D Cparallel C1 C2 Equivalent parallel capacitance sum of capacitances A Q1 V VCD VC VD Qtotal Q1 Q2 CeqV C1V C2V V VAB VA VB Qtotal V V Ceq Capacitors in Series Q Constant Isolated Wire Q Q1 Q2 Constant Q1 Q2 Q Constant VAC VAB VBC Q Q Q Ceq C1 C2 1 Cseries Q1 A 1 1 C1 C2 SERIES Q is same for all capacitors Total potential difference sum of V B C1 Q2 C2 Q Q1 Q2 Ceq C Capacitors in parallel and in series In parallel Cpar C1 C2 Vpar V1 V2 Qpar Q1 Q2 In series 1 Cser 1 C1 1 C2 Vser V1 V2 Qser Q1 Q2 Q1 C1 Qeq Q2 C2 Ceq Q1 Q2 C1 C2 Example Parallel or Series Parallel Circuit Splits Cleanly in Two Constant V What is the charge on each capacitor Qi CiV V 120V Constant Q1 10 F 120V 1200 C Q2 20 F 120V 2400 C Q3 30 F 120V 3600 C Note that Total charge 7200 mC is shared between the 3 capacitors in the ratio C1 C2 C3 i e 1 2 3 C1 10 F C2 20 F C3 30 F 120V Cpar C1 C2 C3 10 20 30 F 60 F Example Parallel or Series Series Isolated Islands Constant Q What is the potential difference across each capacitor Q CserV Q is same for all capacitors Combined Cser is given by 1 1 1 1 Cser 10 F 20 F 30 F Ceq 5 46 F solve above equation Q CeqV 5 46 F 120V 655 C V1 Q C1 655 C 10 F 65 5 V V2 Q C2 655 C 20 F 32 75 V V3 Q C3 655 C 30 F 21 8 V C1 10 F C2 20 F C3 30 F 120V Note 120V is shared in the ratio of INVERSE capacitances i e 1 1 2 1 3 largest C gets smallest V Example Series or Parallel Neither Circuit Compilation Needed In the circuit shown what is the charge on the 10 F capacitor 5 F The two 5 F capacitors are in parallel Replace by 10 F Then we have two 10 F capacitors in series So there is 5V across the 10 F capacitor of interest Hence Q 10 F 5V 50 C 10 F 5 F 10V 10 F 10 F 10V Energy U Stored in a Capacitor Start out with uncharged capacitor Transfer small amount of charge dq from one plate to the other until charge on each plate has magnitude Q dq How much work was needed Q Q 2 2 q Q CV U Vdq dq C 2C 2 0 0 Energy Stored in Electric Field of Capacitor Energy stored in capacitor U Q2 2C CV2 2 View the energy as stored in ELECTRIC FIELD For example parallel plate capacitor Energy DENSITY energy volume u 2 2 Q Q Q 0 Q 0E 2 u 2 2CAd 2 0 A Ad 2 0 A 2 0 A 2 2 2 d volume Ad General expression for any region with vacuum or air Example 10 F capacitor is initially charged to 120V 20 F capacitor is initially uncharged Switch is closed equilibrium is reached How much energy is dissipated in the process Initial charge on 10 F 10 F 120V 1200 C After switch is closed let charges Q1 and Q2 10 F C1 20 F C2 Charge is conserved Q1 Q2 1200 C Q1 400 C Q2 Also Vfinal is same Q1 Q2 Q2 800 C Q1 C1 C2 2 Vfinal Q1 C1 40 V Initial energy stored 1 2 C1Vinitial2 0 5 10 F 120 2 72mJ Final energy stored 1 2 C1 C2 Vfinal2 0 5 30 F 40 2 24mJ Energy lost dissipated 48mJ Summary Any two charged conductors form a capacitor Capacitance C Q V Simple Capacitors Parallel plates C 0 A d Spherical C 4 0 ab b a Cylindrical C 2 0 L ln b a Capacitors in series same charge not necessarily equal potential equivalent capacitance 1 Ceq 1 C1 1 C2 Capacitors in parallel same potential not necessarily same charge equivalent capacitance Ceq C1 C2 Energy in a capacitor U Q2 2C CV2 2 energy density u 0E2 2 Capacitor with a dielectric capacitance increases C C
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