Physics 2102 Gabriela Gonz lez Physics 2102 Capacitors Capacitors and Capacitance Capacitor any two conductors one with charge Q other with charge Q Q Potential DIFFERENCE between Q Uses storing and releasing conductors V Q CV C capacitance Units of capacitance Farad F Coulomb Volt electric charge energy Most electronic capacitors micro Farads mF pico Farads pF 10 12 F New technology compact 1 F capacitors Capacitance Capacitance depends only on GEOMETRICAL factors and on the MATERIAL that separates the two Q conductors Q e g Area of conductors separation whether the space in between is filled We first focus on capacitors with air plastic etc where gap is filled by AIR Electrolytic 1940 70 Electrolytic new Paper 1940 70 Capacitors Variable air mica Tantalum 1980 on Ceramic 1930 on Mica 1930 50 Capacitors and Capacitance Capacitor any two conductors one with charge Q other with charge Q Q Q Potential DIFFERENCE between Uses storing and releasing conductors V Q CV C capacitance Units of capacitance Farad F Coulomb Volt electric charge energy Most electronic capacitors micro Farads mF pico Farads pF 10 12 F New technology compact 1 F capacitors Parallel Plate Capacitor We want capacitance C Q V E field between the plates Gauss Law Q E 0 0A Area of each plate A Separation d charge area s Q A Relate E to potential difference V d Q Qd V E dx dx A 0 A 0 0 0 d What is the capacitance C Q 0A C V d Q Q Parallel Plate Capacitor example A huge parallel plate capacitor consists of two square metal plates of side 50 cm separated by an air gap of 1 mm What is the capacitance C e0A d 8 85 x 10 12 F m 0 25 m2 0 001 m 2 21 x 10 9 F small Lesson difficult to get large values of capacitance without special tricks Isolated Parallel Plate Capacitor Q Q 0 A C V Ed d A parallel plate capacitor of capacitance C is charged using a Q battery Charge Q potential difference V Battery is then disconnected If the plate separation is INCREASED does potential difference V a Increase Q is fixed b Remain the same C decreases e0A d c Decrease Q CV V increases Q Parallel Plate Capacitor Battery Q Q 0 A C V Ed d A parallel plate capacitor of capacitance C is charged using a battery V Charge Q potential difference Plate separation is INCREASED while battery remains connected Does the electric field inside a Increase b Remain the same c Decrease V is fixed by battery C decreases e0A d Q CV Q decreases E Q e0A decreases Q Q Spherical Capacitor What is the electric field inside the capacitor Gauss Law Q E 2 4 0 r Relate E to potential difference between the plates b b Radius of outer plate b Radius of inner plate a Concentric spherical shells Charge Q on inner shell Q on outer shell b kQ kQ V E dr 2 dr r r a a a 1 kQ a 1 b Spherical Capacitor What is the capacitance C Q V Q Q 1 1 4 0 a b 4 0 ab b a Radius of outer plate b Radius of inner plate a Concentric spherical shells Charge Q on inner shell Q on outer shell Isolated sphere let b a C 4 0 a Cylindrical Capacitor What is the electric field in between the plates Radius of outer plate b Radius of inner plate a Length of capacitor L Q on inner rod Q on outer shell Q E 2 0 rL Relate E to potential difference between the plates V E dr cylindrical surface of radius r b b a b Q ln r Q Q b dr ln 2 0 rL 2 0 L a 2 0 L a a Cylindrical Capacitor What is the capacitance C C Q V Q Q b ln 2 0 L a 2 0 L b ln a Radius of outer plate b Radius of inner plate a Length of capacitor L Charge Q on inner rod Q on outer shell Example co axial cable Summary Any two charged conductors form a capacitor Capacitance C Q V Simple Capacitors Parallel plates C e0 A d Spherical C 4p e0 ab b a Cylindrical C 2p e0 L ln b a Capacitors in Parallel A wire is a conductor so it is an equipotential Capacitors in parallel have SAME potential difference but NOT ALWAYS same charge VAB VCD V Qtotal Q1 Q2 CeqV C1V C2V Ceq C1 C2 Equivalent parallel capacitance sum of capacitances PARALLEL V is same for all capacitors Total charge in Ceq sum of charges A C Q1 C1 Q2 C2 Qtotal B D Ceq Capacitors in series Q1 Q2 Q WHY VAC VAB VBC Q1 Q2 B A Q Q Q Ceq C1 C2 1 1 1 Ceq C1 C2 SERIES Q is same for all capacitors Total potential difference in Ceq sum of V C1 C C2 Q Ceq Capacitors in parallel and in series In parallel Ceq C1 C2 Veq V1 V2 Qeq Q1 Q2 C1 Q1 Qeq C2 Q2 Ceq In series 1 Ceq 1 C1 1 C2 Veq V1 V2 Qeq Q1 Q2 Q1 Q2 C1 C2 Example 1 What is the charge on each capacitor Q CV V 120 V Q1 10 mF 120V 1200 mC Q2 20 mF 120V 2400 mC Q3 30 mF 120V 3600 mC Note that Total charge 7200 mC is shared between the 3 capacitors in the ratio C1 C2 C3 i e 1 2 3 10 mF 20 mF 30 mF 120V Example 2 What is the potential difference across each capacitor Q CV Q is same for all capacitors Combined C is given by 10 mF 20 mF 30 mF 1 1 1 1 Ceq 10 F 20 F 30 F 120V Ceq 5 46 mF Q CV 5 46 mF 120V 655 mC Note 120V is shared in the V1 Q C1 655 mC 10 mF 65 5 V ratio of INVERSE V2 Q C2 655 mC 20 mF 32 75 V capacitances i e 1 1 2 V3 Q C3 655 mC 30 mF 21 8 V 1 3 largest C gets smallest V Example 3 10 mF In the circuit shown what is the charge on the 10 F capacitor 5 mF 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 5 mF 10V 10 mF 10 mF 10V Energy 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 q Q CV U Vdq dq C 2C 2 0 0 2 Energy Stored in Electric Field Energy stored in capacitor U Q2 2C CV2 2 View the energy as stored in ELECTRIC FIELD For example …
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