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Capacitance

Capacitance. b. a. L. Capacity to store charge C = Q/V. Q = L l E = l/2pe 0 r V = -( l /2 pe 0 )ln(r/a) C = 2 pe 0 L/ln(b/a). Dimension e 0 x L (F/m) x m. Capacitance. Capacity to store charge C = Q/V. Q = A r s E = r s /e 0 V = Ed C = e 0 A/d. A. d. (F/m) x m.

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Capacitance

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  1. Capacitance b a L Capacity to store charge C = Q/V Q = Ll E= l/2pe0r V = -(l/2pe0)ln(r/a) C = 2pe0L/ln(b/a) Dimension e0 x L (F/m) x m

  2. Capacitance Capacity to store charge C = Q/V Q = Ars E= rs/e0 V = Ed C = e0A/d A d (F/m) x m Increasing area increases Q and decreases C Increasing separation increases V and decreases Q

  3. Capacitance Capacitor microphone – sound vibrations move a diaphragm relative to a fixed plate and change C Tuning  rotate two cylinders and vary degree of overlap with dielectric  change C Changing C changes resonant frequency of RL circuit Increasing area increases Q and decreases C Increasing separation increases V and decreases Q

  4. Increasing C with a dielectric + + - + + + - + + + - + - - - - - - bartleby.com e/e0 = er C  erC To understand this, we need to see how dipoles operate They tend to reduce voltage for a given Q

  5. Point charge in free space Point charge in a medium .E = rv/e0 .E = rv/e0er Effect on Maxwell equations: Reduction of E

  6. Point Dipole R >> d p. R 4pe0R2 V = _________ Note 1/R2 !

  7.  E = -V = -RV/R – (q/R)V/q   p(2Rcosq + qsinq) 4pe0R3 _____________ Point Dipole R >> d p. R 4pe0R2 V = _________ Note 1/R2 !

  8. How do fields create dipoles? - - + + - - + + J J - - + + E - - + + - - + + - - + + - - + + - - + + conductivity Let’s review what happens in a metal A field creates a current density, J=sE, which moves charges to opposite ends, creating an inverse field that completely screens out E

  9. How do fields create dipoles? E - + Charges are not free to move in a dielectric! But electrons can be driven by E a bit away from the Nucleus without completely leaving it, creating an excess Charge on one side and a deficit on the other, .... …. in other words, generating a dipole

  10. Rotating a Dipole + T = (d/2 x F+) - (d/2 x F+) = qd x E = p x E F+ = qE - F- = -qE

  11. That’s how micro-wave ovens work! p = 6.2 x 10-30 Cm 2.5 GHz Radio wave source Absorbed by water, sugar, fats Aligns dipoles built in water molecule and excites atoms “Friction” during rotation in opp. directions causes heating

  12. Polarization and Dielectrics - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + P - + - + - + - + - + - + Instead of creating new dipoles, E could align existing atomic dipoles (say on H20) creating a net polarization P  E

  13. Metal conductor vs Dielectric Insulator Either way, the end result is excess charge on one end and a deficit on the other, like a metal… BUT… There are differences!! In a metal, E forms a current of freely moving charge, and the applied E gets cancelled completely In a dielectric, E creates a polarization of bound but aligned, distorted but immobile charges, and the applied E gets reduced partially (by the dielectric constant).

  14. Metal conductor vs Dielectric Insulator A metal is characterized by a conductivitys which determines its resistance R to current flow, J=sE A dielectric is characterized by a susceptibilityc (and thus a dielectric constant) which determines its capacitance C to store charge, P = Np = e0cE

  15. Dipoles Screen field Displacement vector - + E=(D-P)/e0 D = e0E + P - + -P (opposing polarization Field) - + D (unscreened Field) - + - + - + - + Relative Permittivity er - + P = e0cE D = eE e = e0(1+c) Thus the unscreened external field D gets reduced to a screened E=D/e by the polarizing charges For every free charge creating the D field from a distance, a fraction (1-1/er) bound charges screen D to E=D/e

  16. Free vs Bound Charges - + E=(D-P)/e0 - + -P (opposing polarization Field) - + D (unscreened Field) - + - + - + - + - + e0.E = rtotal =.D- .P = rfree+ rbound

  17. Effect on Capacitance + + - + + + - + + + - + Polarizing charges in dielectric - - - - - - which is why capacitors employ dielectrics (bartleby.com) Free charges on metal e/e0 = K Since the same charge on the plates is now supported by a smaller, screened potential, the capacitance (charge stored by applying unit volt) has actually increased by placement of a dielectric inside! Due to screening, only few of the field lines originating on free charges on the metal plates survive in the Dielectric inside the capacitor

  18.  D.dS = q  E.dl = 0 Differential eqns (Gauss’ law) Fields diverge, but don’t curl Defines Scalar potential E = -U Integral eqns .D = rv  x E = 0 D = eE Constitutive Relation (Thus, E gets reduced by er) er = 1 (vac), 4 (SiO2), 12 (Si), 80 (H20) ~2 (paper), 3 (soil, amber), 6 (mica), Effect on Maxwell equations: Reduction of E

  19. Point charge in free space Point charge in a medium .E = rv/e0 .E = rv/e0er

  20. Electrostatic Boundary values Supplement with constitutive relation D=eE  D.dS=q .D = r  x E = 0  E.dl = 0 Maxwell equations for E D1n rs D2n Use Gauss’ law for a short cylinder Only caps matter (edges are short!) D1n-D2n = rs Perpendicular D discontinuous

  21. Electrostatic Boundary values Supplement with constitutive relation D=eE  D.dS=q .D = r  x E = 0  E.dl = 0 Maxwell equations for E E1t E2t No net circulation on small loop Only long edges matter (heights are short!) E1t-E2t = 0 Parallel E continuous

  22. Electrostatic Boundary values Perpendicular D discontinuous Parallel E continuous Can use this to figure out bending of E at an interface (like light bending in a prism)

  23. Example: Bending e1E1n=e2E2n Parallel E continuous Perpendicular D continuous if no free charge rs at interface e2 e1 q1 q2 E1t=E2t cosq1=cosq2 tanq1/tanq2 = e2/e1 D1n=D2n e1sinq1=e2sinq2 e2 > e1 means q2 < q1

  24. Conductors are equipotentials • Conductor  Static Field inside zero (perfect screening) • Since field is zero, potential is constant all over • Et continuity equation at surface implies no field • component parallel to surface • Only Dn, given by rs.

  25. Field lines near a conductor + + + - - - - + - + - - - + + + Equipotentials bunch up here  Dense field lines Principle of operation of a lightning conductor Plot potential, field lines

  26. Images So can model as Charge + Image Compare with field of a Dipole! Charge above Ground plane (fields perp. to surface) Equipotential on metal enforced by the image

  27. Images

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