Prozorov_05 - PHYSICS 222 Introduction to Classical Physics...

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Unformatted text preview: PHYSICS 222 Introduction to Classical Physics II Prof. Ruslan Prozorov Iowa State University Fall 2011 LECTURE 5 Capacitance and capacitors. Dielectrics. Electric field energy. keep charges apart and you get capacitance PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 2 capacitance remember from previous lecture: V d 0 Q A Capacitance is defined as: C - only geometry! 0 V d PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 3 capacitor With the same potential difference store more charge if capacitance is larger A Q 0 V C V d PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 4 capacitors of different geometry: sphere use the definition: Q A C 0 V d look at the Gaussian surface between the shells. From Gauss’s law, electric field between the shells is: 1 Q E r 4 0 r 2 Electric potential between the shells is: Q dr Q 1 1 V Edr r 2 4 0 ra rb 4 0 ra ra rb rb PHYS222 - Prof. Ruslan Prozorov - Iowa State University so, the capacitance is: ra rb Q C 4 0 V rb ra 31 August 2011 5 cylindrical capacitor 1 E 2 0 r rb dr Q V Edr r 2 0 L ln ra 2 0 r r rb a so, the capacitance is: rb a 2 0 Q C V ln rb ra PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 6 d the microphone Current sensor Current sensor Battery Battery (fixed potential difference) Moveable plate Fixed plate Sound waves incident pressure oscillations oscillating plate separation d oscillating capacitance (C ~ 1/d) oscillating charge on plate (Q ~ C) oscillating current in wire (I = dQ/dt) oscillating electrical signal in sensor PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 7 real capacitors PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 8 very large capacitors capacitors are different from batteries: capacitors store charge, whereas battery maintains voltage only up to a certain current (due to internal resistance). The charge can be discharged very quickly reasulting in huge currents. The array above right can store up to 50 MegaJouls (~ 10 kg TNT) PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 9 how to increase capacitance? what if we put a dielectric in between the plates? The applied voltage will still be below the electric breakthrough. Dielectric polarizes, so the effective electric field reduces. Now, let’s charge a capacitor, so it has charge Q on each plate. Then, we insert a dielectric. The charge is conserved, so: dielectric constant Cempty Q Q Cdielectric Vempty Vdielectric PHYS222 - Prof. Ruslan Prozorov - Iowa State University Cdielectric K 1 Cempty 31 August 2011 10 molecular models PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 11 polarization and electric field lines PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 12 dielectric constants note that there are no perfect insulators. These is always some leakage electric current, so that the capacitor slowly discharges internally! Only perfect vacuum would work. PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 13 Gauss’s Law in dielectrics electric field is reduced due to induced surface charge PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 14 in terms of electric field and surface charge The polarization produces an induced charge on the surfaces of the dielectric that are in contact with the capacitor plates. E E 0 E polar i 0 E0 E K K 0 i K PHYS222 - Prof. Ruslan Prozorov - Iowa State University 1 i 1 K 31 August 2011 15 dielectric breakdown PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 16 permittivity permittivity is defined as: K 0 for example, electric field of a point charge inside a dielectric will be obtained by replacing 1 Q E r 4 0 r 2 with 0 1 Q E r 4 r 2 is called permittivity of vacuum PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 17 capacitors in series E0 d E1 , V1 E1 K1 2 Q Q 1 1 1 V V1 V2 E0 d C1 C2 C C1 C2 E2 , V2 E2 K2 2 PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 18 capacitors in parallel Q1 Q2 Q V V1 V2 C C1 C2 C1 C2 C PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 19 calculating capacitance do it in parts 1 1 3 1 1 12 6 12 4 C C 4 F C 3 11 4 18 F PHYS222 - Prof. Ruslan Prozorov - Iowa State University 1 1 3 1 1 18 9 18 6 C C 6 F 31 August 2011 20 energy stored in capacitors The work required to bring small charge dQ from one plate to another is: Q dW VdQ dQ C so, the total work to charge a capacitor from Q=0 to some Q is: W Q 0 Q Q2 dQ C 2C this work is (ideally) converted into a potential energy (similar to when we lift a weight to some heights). This energy is now stored in a capacitor. Q 2 QV CV 2 U 2C 2 2 PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 21 size of a capacitor to store 50 MJ Suppose the capacitor operates at 10 V 2U 2 50 106 C 2 106 F V 102 CV 2 U 2 A C d is it large? take our “best” dielectric from the earlier tables – strontium titanate with K = 310 and spacing between the plates of 1 mm. Then the area will be: 106 103 A 3.2 m 2 310 Cd charge? Q CV 107 C without dielectric we would need 1000 m2 area PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 22 energy of an electric field The work done to charge a capacitor was done against an electric field d dW EdQ 0 so, we may think of electric field energy. It is usually expressed as energy density: U CV 2 1 u E2 Volume 2 Ad u 2 A C , V Ed d PHYS222 - Prof. Ruslan Prozorov - Iowa State University 31 August 2011 23 ...
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This note was uploaded on 11/14/2011 for the course PHYS 5863005 taught by Professor Meyer during the Fall '09 term at Iowa State.

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