Physics 2102 Jonathan Dowling Physics 2102 Lecture 08 THU 18 FEB Capacitance II 25 4 7 QuickTime and a decompressor are needed to see this picture 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 Qtotal Q1 Q2 CeqV C1V C2V V VAB VA VB A C Q1 C1 VA VB Q2 C2 VC VD B D V VCD VC VD Ceq C1 C2 Equivalent parallel capacitance sum of capacitances Cparallel C1 C2 PAR V Parallel V the Same Qtotal Ceq V V Q Constant Q1 Q2 Q Constant VAC VAB VBC Isolated Wire Q Q1 Q2 Constant Q1 SERI Q Series Q the Same Q Q Q Ceq C1 C2 1 Cseries 1 1 C1 C2 Q2 B A C1 C C2 Q Q1 Q2 SERIES Q is same for all capacitors Total potential difference sum of V Ceq 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 Series Parallel Circuit Splits Cleanly in Two Constant V What is the charge on each capacitor Qi C i V V 120V on ALL Capacitors PAR V Q1 10 F 120V 1200 C Q2 20 F 120V 2400 C Q3 30 F 120V 3600 C Note that Total charge 7200 C 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 Series Series Isolated Islands Constant Q What is the potential difference across each capacitor C1 10 F Q CserV Q is same for all capacitors SERI Q Combined Cser is given by 1 1 1 1 Cser 10 F 20 F 30 F C2 20 F C3 30 F 120V Ceq 5 46 F solve above equation Q CeqV 5 46 F 120V 655 C V1 Q C1 655 C 10 F 65 5 Note 120V is shared in the ratio of INVERSE capacitances i e 1 1 2 1 3 largest C gets smallest V 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 by symmetry 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 How much work was needed Q Q dq 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 Q Q Q 0 Q 0 E 2 u 2 2CAd 2 0 A Ad 2 0 A 2 0 A 2 2 2 2 d volume Ad General expression for any region with vacuum or air Dielectric Constant If the space between capacitor DIELECTRIC plates is filled by a dielectric the capacitance INCREASES by a factor This is a useful working definition for dielectric constant Typical values of are 10 200 Q Q C A d The and the constant o are both called dielectric constants The has no units Atomic View Emol Molecules set up counter E field Emol that somewhat cancels out capacitor field Ecap This avoids sparking dielectric breakdown by keeping field inside dielectric small Ecap Hence the bigger the dielectric constant Example Capacitor has charge Q voltage V Battery remains connected while dielectric slab is inserted Do the following increase decrease or stay the same Potential difference Capacitance Charge Electric field dielectric slab Example Initial values capacitance C charge Q potential difference V electric field E Battery remains connected V is FIXED Vnew V same Cnew C increases Qnew C V Q increases dielectric slab Since Vnew V Enew V d E same Energy stored u 0E2 2 u 0E2 2 E2 2 Summary Any two charged conductors form a capacitor Capacitance C Q V Simple Capacitors Parallel plates C 0 A d 0 ab b a Cylindrical C 2 0 L ln b a Spherical C 4 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 E2 2 Capacitor with a dielectric capacitance increases C kC
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