General Physics L14_capacitance A device storing electrical energy The capacitor A device storing electrical energy §18.3

Capacitor A potential across connected plates causes charge migration until equilibrium Charge stored q = CDV C = capacitance Unit = C/V = farad= F DV – – – + + + –q +q

Capacitance Amount of charge separation per volt A large capacitance means it doesn’t take much voltage to separate a lot of charge “squishy”

The Farad Very large unit Circuit capacitors typically in pF or nF range Some modern capacitors have capacitances ≈ 1F, but typically allow only small voltages across

General Physics L14_capacitance At Equilibrium Capacitor charges to potential DV Capacitor charge Q = CDV + – DV DV C + –

General Physics L14_capacitance Capacitor Charge A charged capacitor stores energy “in the field” Potential energy of separated charges Can discharge very quickly C DV +            – + –

Field Around Infinite Plate With uniform charge density s = q/A s e0 1 2 E = e0 = 8.8510–12 C2 N m2

Infinite ||-Plate capacitor Individually Together –q 1/2 s/e0 +q −q s/e0 +q 1/2 s/e0

General Physics L14_capacitance Finite Capacitor Uniform E field between plates Small “fringe” field at edges – + E Fields cancel outside d

Capacitance of a Capacitor Behavior from design §19.5

General Physics L14_capacitance Finite Capacitor Parallel plates of opposite charge Charge density s = Q/A – + Fields cancel outside s/e0 Potential DV = dE = d s/e0 = d Q/(Ae0) Capacitance C = Q/V = e0 A/d d

Parallel Plate Capacitance Plate area A, plate separation d d A Field E = s e0 = Q Ae0 Potential DV = Ed = Qd Ae0 Capacitance Q/DV = Q Ae0 Qd Ae0 d =

Circuit Element Symbols + – DV Potential Source Conductor Capacitor or Resistor

Energy in a Capacitor C = Q/V so V = Q/C It takes work to push charge Q Voltage to push DQ proportional to Q V Q slope = 1/C

Energy in a Capacitor Work to charge to Q is area of triangle W = 1/2 Q(Q/C) = 1/2 Q2/C Work to charge to V W = 1/2 V (CV) = 1/2 C(V)2 V Q Q/C CDV

Combining Capacitors Parallel and Series

Parallel Capacitors All have the same potential difference Capacitances add (conceptually add A’s)

Series Capacitors All have the same charge separation Reciprocals of C are additive (conceptually add d’s)

Capacitor with a Dielectric Fill the space between the plates with a polarizable insulator (dielectric) Reduces E field between plates Stabilizes charge on plates + + + + + + + + + + − − − − − − − − − − + −

Capacitor with a Dielectric If capacitance without dielectric is C, capacitance with dielectric is kC. k = dielectric constant k

Dielectric Insulator Polarizes in field Effectively reduces plate separation d Reduces field between plates Dielectric constant = relative permittivity e = ke0 Capacitance C = Ae/d

Dielectric breakdown Strong field can separate charges Ejects electrons from their orbitals Dielectric becomes a conductor Damage usually permanent Limits practical thinness of dielectric layer

Some Dielectrics Material k Strength (kV/mm) Air 1.0006 3 Paper 3.85 16 Teflon 2.1 19.7 Mica 3–6 118 TiO2 86–173 4 Silica 470–670

RC Circuits resistor + capacitor §20.13

Charging a Capacitor e Initial charge Q = 0 Close switch at t = 0 + – R Initial charge Q = 0 Close switch at t = 0 Uncharged capacitor acts as a conductor Charged capacitor acts as a break in the circuit Q = Ce (1 − e−t/t) I = e /R e−t/t

Time Constant t Characteristic time of the circuit At current e/R, time to transfer charge Q Value t = RC Units = WF = s

Discharging a Capacitor +Q0 −Q0 Initial charge Q = Q0 Initial voltage V0 = Q0/C Close switch at t = 0 C R Q = Q0 e−t/t I = V0/R e−t/t