hapter 4: apacitance and Dielectrics apacitor: two conductors (separated by an insulator) usually oppositely charged a b - ab proportional to charge / ab (defines capacitance) units: F / pc4: The parallel plate capacitor A E σ ε 0 A ε 0 d - A ab Ed d Aε 0 A ε 0 ab d apacitance does not depend upon,! > depends upon geometric factors only How big is Farad? (parallel plate example) pc4:
Typical apacitances ~ µf, nf, pf Example: A parallel plate capacitor has plates.00 m in area, separated by a distance of 5.00 mm. A potential difference of 0,000 is applied across the capacitor. Determine the capacitance the charge on each plate the magnitude of the electric field in the region between the plates. pc4: 3 A long cylindrical capacitor ab λ ln r b πε 0 λl πε L 0 lnr b L πε 0 lnr b coax: 70 pf m r b L pc4: 4
A long cylindrical capacitor, small distance between cylinder walls πε L 0 lnr b r b d πε 0 L ln d r b R >> d [( ) ] πε L 0 ln( x) x x x 3 3 L πε L 0 d R πrlε 0 d ln d R A d ε 0 r b L apacitor looks approximately like parallel plates, in appropriate limit. pc4: 5 apacitors in circuits symbols analysis follow from conservation of energy (in terms of electric potential) conservation of charge pc4: 6
pc4: 7 apacitors in series ab a c b eq eq A 3 µf capacitond a 6 µf capacitore connected in series across an 8 battery. Determine the equivalent capacitance, the charge on each capacitond the potential difference across each capacitor. pc4: 8 apacitors in parallel eq eq A 3 µf capacitond a 6 µf capacitore connected in parallel across an 8 battery. Determine the equivalent capacitance, the potential difference across each capacitond the charge on each capacitor. ab a b
ombinations of combinations can be analyzed piecewise 3 Some configurations are not combinations that can be treated as combinations that can be analyzed as serial/parallel 5 3 4 pc4: 9 Energy stored in a capacitor When charged: harging q v dw dqv dq W q q 0 dq q q U dq q -q -dq v q/ dq q -q dq U pc4: 0
Electric Field Energy Uniform field: parallel plate capacitor U ε A 0 volume Ad Ed d u U / volume energy density ε 0 A d (Ed) /(Ad) u ε 0 E pc4: In the circuit shown 48, 9µF, 4µF and 3 8µF. (a)determine the equivalent capacitance of the circuit, (b) determine the energy stored in the combination by calculating the energy stored in the equivalent capacitance, (c) calculate the charge on and potential difference across each capacitond (d) calculate the energy stored in each individual physical capacitor. 3 pc4:
Dielectrics: insulating materials with other interesting properties In parallel plate capacitors Fo charged, isolated capacitor 0 potential difference decreases same charge > capacitance increases / > 0 / 0 Dielectric onstant: K / 0 material property pc4: 3 Effect of dielectric on Electric field parallel plates, constant charge 0 0 > 0 / (reduced) > E E 0 /K Material is polarized Effective surface charge distribution E 0 σ ε 0 ε Kε 0 E σ net σ σ i σ i σ ε 0 ε 0 K permittivity of dielectric σ σ σ i σ i E σ ε Kε 0 A d ε A d u Kε 0E εe σ net σ σ i σ net (σ σ i ) pc4: 4
Example 5-8: Take a parallel plate capacitor whose plates have an area of 000 cm and are separated by a distance of cm. The capacitor is charged to an initial voltage of 3 k and then disconnected from the charging source. An insulating material is placed between the plates, completely filling the space, resulting in a decrease in the capacitors voltage to k. Determine the original and new capacitance, the charge on the capacitor, the dielectric constant of the material, the permittivity of the dielectric, the original and new electric fields, the energy stored in the capacitor with and without the dielectric. pc4: 5 How does an insulating dielectric material reduce electric fields by producing effective surface charge densities? Reorientation of polar molecules Induced polarization of non-polar molecules Dielectric Breakdown: breaking of molecular bonds/ionization of molecules. pc4: 6
Polarization (approximately) proportional to applied Electric Field beyond lineapproximation: nonlinear optics... Dielectric materials and Gauss s Law r r KE da ε 0 r r εe da enclosed enclosed r r D da enclosed enclosed free charge r r D εe Electric Displacement ( trivia) pc4: 7