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Download Chapter 24: Capacitance and Dielectrics and Ch. 26
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Chapters 24 and 26.4-26.5 1 Capacitor ++++++++++ ++++++++++++++ +q Potential difference=V -q ------------------------- Any two conductors separated by either an insulator or vacuum for a capacitor The “charge of a capacitor” is the absolute value of the charge on one of conductors. q constant This constant is called the “capacitance” and V is geometry dependent. It is the “capacity” for holding charge at a constant voltage 2 Units 1 Farad=1 F= 1 C/V Symbol: Indicates positive potential 3 Interesting Fact When a capacitor has reached full charge, q, then it is often useful to think of the capacitor as a battery which supplies EMF to the circuit. 4 Simple Circuit +q H -q L Initially, After S is H closed, & L =0 H=+q L=-q S i -i 5 Recalling Displacement Current Maxwell thought of the capacitor as a flow device, like a resistor so a “displacement current” would flow between the plates of the capacitor like this -q +q i i id This plate induces a negative charge here Which means the positive charge carriers are moving here and thus a positive current moving to the right 6 If conductors had area, A Then current density would be Jd=id/A 7 Calculating Capacitance Calculate the E-field in terms of charge and geometrical conditions Calculate the voltage by integrating the E-field. You now have V=q*something and since q=CV then 1/something=capacitance 8 Parallel plates of area A and distance, D, apart q E dA Area, A 0 EA q 0 E q A 0 V E ds E ds ED f i q V ED and E A 0 V ++++++++++ ++++++++++++++ Distance=D ------------------------- qD q A 0 C A 0 1 C qD D A 0 9 Coaxial Cable—Inner conductor of radius a and thin outer conductor radius b q E dA 0 E 2rL q 0 E q 2rL 0 b V E ds E drrˆ f i q dr 2L 0 r a q b ln 2L 0 a 2L 0 C 1 1 b b ln ln 2L 0 a a C 20 L b ln a V -q +q 10 Spherical Conductor—Inner conductor radius A and thin outer conductor of radius B q E dA 0 q E 4r E 0 4r 2 0 2 q B V E ds E drrˆ f i q dr 2 4 r 0 A q 1 1 q B A V 40 B A 40 AB 1 AB 40 C 1 B A B A 40 AB 11 Isolated Sphere of radius A q 1 1 V Let B 40 B A q V 40 A C 40 A 12 Capacitors in Parallel i1 i E i2 i3 E C1 q1 C2 q2 C3 q3 i=i1+i2+i3 implies q=q1+q2+q3 i Ceq E q CeqV C1V C2V C3V Ceq C1 C2 C3 13 Capacitors in Series i i C1 By the loop rule, E=V1+V2+V3 q E E C2 q E q Ceq C3 q E q q q C1 C2 C3 Ceq q q q q Ceq C1 C2 C3 1 1 1 1 Ceq C1 C2 C3 14 Energy Stored in Capacitors d Work V d charge Work Vdq q And V C q 1 2 Work dq q C 2C Or 1 Work CV 2 2 Technically, this is the potential to do work or potential energy, U U=1/2 CV2 or U=1/2 q2/C Recall Spring’s Potential Energy U=1/2 kx2 15 Energy Density, u u=energy/volume Assume parallel plates at right Vol=AD U=1/2 CV2 C 0 A D 1 0 A 2 u V 2 AD D 1 V2 1 u 0 2 0E2 2 D 2 Area, A ++++++++++ ++++++++++++++ Distance=D ------------------------- Volume wherein energy resides 16 Dielectrics Area, A ++++++++++ ++++++++++ ++++++++++ ++++++++++++++ ++++++++++++++ ++++++++++++++ --------------------------------------------------------------------------- Insulator Distance=D Voltage at which the insulating material allows current flow (“break down”) is called the breakdown voltage 1 cm of dry air has a breakdown voltage of 30 kV (wet air less) 17 The capacitance is said to increase because we can put more voltage (or charge) on the capacitor before breakdown. The “dielectric strength” of vacuum is 1 Dry air is 1.00059 So we can replace, our old capacitance, Cair, by a capacitance based on the dielectric strength, k, which is Cnew=k*Cair An example is the white dielectric material in coaxial cable, typically polyethylene (k=2.25) or polyurethane (k=3.4) Dielectric strength is dependent on the frequency of the electric field 18 Induced Charge and Polarization in Dielectrics E ++++++++++++++++++++++ Note that the charges have separated or polarized -- - - - - - - - - - - - - - - - - Ei +++++++++++++++++++ ----------------------- E0 k E0 0 E0 0 i k 0 0 1 i 1 k ETotal=E0-Ei 19 Permittivity of the Dielectric k0 For real materials, we define a “D-field” where D=k0E D da q freeenclosed For these same H ds i freeenclosed D materials, there can be a t magnetization based on D D da the magnetic susceptibility, c, : H= cm0B 20 Capacitor Rule For a move through a capacitor in the direction of current, the change in potential is –q/C If the move opposes the current then the change in potential is +q/C. move Vaa-Vbb= -q/C +q/C Va i Vb 21 RC Circuits Initially, S is open so at t=0, i=0 in the resistor, and the charge on the capacitor is 0. Recall that i=dq/dt R A S B V C 22 Switch to A Start at S (loop clockwise) and use the loop rule R A S B q iR V 0 C q V iR C dq q V R dt C V C 23 An Asatz—A guess of the solution My ansatz : q p A Be t RC t dq p B RC e dt RC dq q V R dt C B A Be V e C C at t 0, q 0 t RC q p 0 A Be 0 RC t RC A V or A CV C 0 CV B 0 B CV q (t ) CV (1 e t RC ) 24 Ramifications of Charge At t=0, q(0)=CV-CV=0 At t=∞, q=CV (indicating fully charged) What is the current between t=0 and the time when the capacitor is fully charged? t d d i q (t ) CV 1 e RC dt dt t t CV RC V RC i e e RC R 25 Ramifications of Current At t=0, i(0)=V/R (indicates full current) At t=∞, i=0 which indicates that the current has stopped flowing. Another interpretation is that the capacitor has an EMF =V and thus R A S Circuit after a very long time B V ~V 26 Voltage across the resistor and capacitor Potential across resistor, VR V VR iR e RC Ve RC R t t R A S Potential across capacitor, VC B V C t RC CV 1 e q VC C C t VC V 1 e RC At t=0, VC=0 and VR=V At t=∞, VC=V and VR=0 27 RC—Not just a cola RC is called the “time constant” of the circuit RC has units of time (seconds) and represents the time it takes for the charge in the capacitor to reach 63% of its maximum value When RC=t, then the exponent is -1 or e-1 t=RC 28 Switch to B The capacitor is fully charged to V or q=CV at t=0 S CV B q iR 0 V C dq q R dt C dq 1 t dt ln q k q RC RC q (t ) Ae t RC C t and R A A RC i (t ) e RC If q CV at t 0, q(0) Ae0 A CV 29 Ramifications At t=0, q=CV and i=-V/R At t=∞, q=0 and i=0 (fully discharging) Where does the charge go? The charge is lost through the resistor 30 Three Connection Conventions For Schematic Drawings Connection Between Wires A B C No Connection 31 Ground Connectors Equivalently 32 Household Wiring “hot” or black “return”/ “neutral” or white “ground” or green Single Phase Rated 20 A (NW-14) Max V 120 VAC Normally, the “return” should be at 0 V w.r.t. ground In THEORY, but sometimes no! 33 The Death of Little Johnny A short develops between the hot lead and the washer case hot Little Johnny X X Washer Uhoh! It leaks! neutral RG RG=∞, =0, then IfIf R then G Johnny is dead! Johnny is safe 120V RLittle Johnny RG 34 Saving Little Johnny A short develops between the hot lead and the washer case Little Johnny hot Washer Uhoh! It leaks! neutral RG No Path to Johnny! 120V RLittle Johnny RG 35