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Relation between field and potential E = -V 2 ∫E.dl = -(V2-V1) along any path 1 • Set reference point 1 at infinity where V1 = 0 • Use linear path from infinity to desired point 2, coordinate r r V(r) = -∫E.dl ∞ 1 Equation satisfied by potential xE=0 . E = r/e0 E = -V 2V = - r/e0 Poisson’s equation Laplace’s equation Free space 2V = 0 2 Continuous distribution of charges E= rv(R’) R’ ________ 4pe0|R’|3 d3 R ’ R‘ rv(R’) V = ________ d3 R ’ 4pe0|R’| 3 Various examples Charged Disc V=? Charged line Charged Sheet Charged Ball Charged Hollow Ball Calculate V directly, or from E obtained through Gauss’ Law What do equipotentials and field lines look like? 4 Equipotentials Familiar examples Equipotentials Connect pts. with same V E = -V runs perpendicular to it Point charge 5 How to draw equipotentials? Point Dipole R >> d p. R V = _________ 4pe0R2 Note 1/R2 ! 7 Point Dipole p. R V = _________ R >> d 4pe0R2 Note 1/R2 ! E = -V = -RV/R – (q/R)V/q p(2Rcosq + qsinq) _____________ 4pe0R3 8 So dipoles annihilate each other, thereby countering the field that separated and created them in the first place. In other words, they conspire to produce a polarizing field. 9 Electric Field Applet http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html9/ 30/ 12 10:37 Electric Field Applet Done. 1 Go Random ize Ref resh Reset Potential Gradient Equipotential Lines Electric Field Lines Initial Velocity Wall Boundary System Charge: - 13.0 10 Electric Field Applet http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html9 / 30/ 12 10:37 P Electric Field Applet Done. 1 Go Random ize Ref resh Reset Potential Gradient Equipotential Lines Electric Field Lines Initial Velocity Wall Boundary System Charge: - 13.0 11 Conductors are equipotentials • Conductor Static Field inside zero (perfect screening) • Since field is zero, potential is constant all over • E is perpendicular to the conducting surface 12 Images Charge above Ground plane (fields perp. to surface) Compare with field of a Dipole! Equipotential on metal enforced by the image So can model as Charge + Image 13 Field lines near a conductor + + + --+ -- -- + + + + Equipotentials bunch up here Dense field lines Principle of operation of a lightning conductor Plot potential, field lines 14 How much charge can we store on a metal? We can calculate the voltage on a metal for a given charge. Conversely, we can calculate the charge we need to store to create a given voltage on a metal. How would we quantify the charge that is needed to create 1 volt on a metal? The ‘Capacitance’ 15 Capacitance Capacity to store charge C = Q/V a b L Q = Ll E = l/2pe0r V = -(l/2pe0)ln(r/a) C = 2pe0L/ln(b/a) Dimension e0 x L (F/m) x m 16 Capacitance Capacity to store charge C = Q/V A d Q = Ars E = rs/e0 V = Ed C = e0A/d (F/m) x m Increasing area increases Q and decreases C Increasing separation increases V and decreases Q 17 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 18 Increasing C with a dielectric + + + - - + - + + - e/e0 = er C erC + - + + - bartleby.com To understand this, we need to see how dipoles operate They tend to reduce voltage for a given Q 19 Dipoles Screen field + + - -P + - (opposing + -polarization+ - Field) + + + E=(D-P)/e0 D (unscreened Field) 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 20 Effect on Maxwell equations: Reduction of E Point charge in free space .E = rv/e0 Point charge in a medium .E = rv/e0er 21