Electrostatics: Capacitor Examples
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1 Electrostatics: apacitor Examples EE3321 Electromagnetic Fiel Theory Outline Parallel plate capacitor How big is a Fara? oaxial capacitor RG-59 coax Inhomogeneous capacitor Electrostatics -- apacitor Examples lie 2 1
2 Example #1: Parallel Plate apacitor tep 1 hoose a convenient coorinate system. artesian Electrostatics -- apacitor Examples lie 4 2
3 tep 1 hoose a convenient coorinate system. artesian tep 2 et the plates carry charges + an. Electrostatics -- apacitor Examples lie 5 tep 3 alculate D using Gauss law. Recall the fiel aroun an infinite plate. D aˆ Fiel below the top plate, D aˆ top 2 n 2 z D top Electrostatics -- apacitor Examples lie 6 3
4 tep 3 alculate D using Gauss law. Recall the fiel aroun an infinite plate. D aˆ Fiel below the top plate, D aˆ top bot 2 n 2 n Fiel above the bottom plate, D aˆ 2 z D bot Electrostatics -- apacitor Examples lie 7 tep 3 alculate D using Gauss law. Recall the fiel aroun an infinite plate. D aˆ Fiel below the top plate, D aˆ top bot 2 n 2 n Fiel above the bottom plate, D aˆ 2 n When both plates are consiere D D D a top bot ˆz D Electrostatics -- apacitor Examples lie 8 top D bot 4
5 tep 3 alculate D using Gauss law. The surface charge ensity is The final express for D is ˆ D ˆ az az D Electrostatics -- apacitor Examples lie 9 tep 4 alculate E. We calculate E from the constitutive relation. D E a ˆz E Electrostatics -- apacitor Examples lie 1 5
6 tep 5 alculate V. Given E, we calculate V by integrating from the bottom plate to the top plate. V E V ˆ ˆ azza V V z z Electrostatics -- apacitor Examples lie 11 tep 6 alculate capacitance. V The final answer is elf-check shoul not be a function of or V. Electrostatics -- apacitor Examples lie 12 6
7 Example #2: How Big is a Fara? Example #2 uppose our plates are 1 m by 2 m an the gap between the plates is 1 mm. W r F F m1.1 m2 m.1 m 1.78 μf Our capacitor is physically very large, yet the capacitance is very small. The Fara is a HUGE unit!!! Electrostatics -- apacitor Examples lie 14 7
8 Example #3: oaxial apacitor Example #3 oaxial apacitor tep 1 hoose a convenient coorinate system. ylinrical (,,z) Electrostatics -- apacitor Examples lie 16 8
9 Example #3 oaxial apacitor tep 1 hoose a convenient coorinate system. ylinrical (,,z) tep 2 et the plates carry charges + an. Electrostatics -- apacitor Examples lie 17 Example #3 oaxial apacitor tep 3 alculate D using Gauss law. We efine a Gaussian surface with raius to be insie of the ielectric. Ds From our bounary conitions, we know the electric fiel will be normal at the interfaces to the metal. Electrostatics -- apacitor Examples lie 18 9
10 Example #3 oaxial apacitor tep 3 alculate D using Gauss law. The only fiel configuration that makes sense consiering the bounary conitions is when the fiel is purely raially irecte. D D,, zaˆ Electrostatics -- apacitor Examples lie 19 Example #3 oaxial apacitor tep 3 alculate D using Gauss law. Gauss law becomes 2 ˆ ˆ ˆ ˆ D a za za za z 2 D z 2 D 2 D z z 2D z 2 D Electrostatics -- apacitor Examples lie 2 1
11 Example #3 oaxial apacitor tep 3 alculate D using Gauss law. olving for D gives D,, z 2 D aˆ 2 tep 4 alculate E. We calculate E from the constitutive relation. D E a ˆ 2 D only epens on. Electrostatics -- apacitor Examples lie 21 Example #3 oaxial apacitor tep 5 alculate V. Given E, we calculate V by integrating from the inner conuctor to the outer conuctor. V E b a b a ˆ a ˆ 2 2 a b 1 2 b ln a a 2 Electrostatics -- apacitor Examples lie 22 11
12 Example #3 oaxial apacitor tep 5 alculate V. ontinue b V ln a ln bln a ln aln b a ln 2 b Electrostatics -- apacitor Examples lie 23 Example #3 oaxial apacitor tep 6 alculate capacitance. V a ln 2 b 2 a ln b * elf heck is not a function of or V. Electrostatics -- apacitor Examples lie 24 12
13 Example #3 oaxial apacitor Distribute apacitance We woul like to specify the capacitance without knowlege of. We o this using the istribute capacitance, which capacitance per unit length. 2 a ln b Electrostatics -- apacitor Examples lie 25 Example #4: RG-59 oax 13
14 Example #4 RG-59 oax A stanar RG-59 coax has Inner conuctor iameter: Outer conuctor iameter: Dielectric constant: pecifie capacitance:.81 mm (2 AWG) 3.66 mm pf/m 2r ln F m 2.1 ab 12 ln 3.66 mm.81 mm F m pf m Electrostatics -- apacitor Examples lie 27 Example #5: Inhomogeneous apacitor 14
15 Example #5 Inhomogeneous apacitor uppose we have an inhomogeneous capacitor. Electrostatics -- apacitor Examples lie 29 Example #5 Inhomogeneous apacitor We split the inhomogeneous capacitor into a combination of homogeneous capacitors. Electrostatics -- apacitor Examples lie 3 15
16 Example #5 Inhomogeneous apacitor alculate each homogeneous capacitor inepenently. tw Electrostatics -- apacitor Examples lie 31 Example #5 Inhomogeneous apacitor alculate each homogeneous capacitor inepenently. tw tw 1 Electrostatics -- apacitor Examples lie 32 16
17 Example #5 Inhomogeneous apacitor alculate each homogeneous capacitor inepenently. tw tw t w w Electrostatics -- apacitor Examples lie 33 Example #5 Inhomogeneous apacitor We view our capacitor as a series/parallel combination of capacitors. The equivalent capacitance is eq tw1 t w w 2tw eq t31w12w2 w1w2 w w w w Electrostatics -- apacitor Examples lie 34 17
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