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Potential Difference: Path Independence

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  1. Potential Difference: Path Independence f i Path independence principle: V between two points does not depend on integration path

  2. Potential Difference in Metal f i In static equilibrium E = 0 What is E inside metal? In static equilibrium the electric field is zero at all locations along any path through a metal. What is the potential difference (Vf – Vi)? The potential difference is zero between any two locations inside the metal, and the potential at any location must be the same as the potential at any other location. Is V zero everywhere inside a metal? No! But it is constant

  3. Potential in Metal -Q +Q E d =3 mm In static equilibrium A Capacitor with large plates and a small gap of 3 mm has a potential difference of 6 Volts from one plate to the other. -3 V +3 V Charges are on surface V = 6 Volt

  4. Potential in Metal -Q1 +Q1 d =3 mm In static equilibrium Insert a 1 mm thick metal slab into the center of the capacitor. 1 mm Metal slab polarizes and has charges +Q2 and -Q2 on its surfaces. What are the charges Q1 and Q2? X At X E=0 inside metal Q2=Q1 Now we have 2 capacitors instead of one Ignoring the fringe fields, E = 2000V/m in each capacitor (from previous slide). V = 4 V V inside metal slab is zero! Charges +Q2 and –Q2 There is no “conservation of potential”!

  5. Potential in Metal Nonzero electric field of uniform magnitude E throughout the interior of a wire of length L. Direction of the field follows the direction of the wire. There can be a potential in metal if is NOT in static equilibrium • Metal is not in static equilibrium: • When it is in the process of being polarized • When there is an external source of mobile charges (battery) For each step , the potential difference is:V = -EL If a metal is not in static equilibrium, the potential isn’t constant in the metal.

  6. Question 300 V/m 0 V/m 300 V/m A B 0.04m 0.02m 0.03m 270 V -270 V -18 V 6 V -6 V What is VB-VA?

  7. Question 0 V/m A B 0.04m 0.02m 0.03m x 0 =

  8. Round Trip Potential Difference + Potential difference due to a stationary point charge is independent of the path Potential difference along a closed loop is zero

  9. Predicting Possible Field Configuration is always parallel to Is the following “curly” pattern of electric field possible? dl dl dl This “curly” pattern of electric field is impossible to produce by arranging any number of stationary point charges!

  10. Potential of a Uniformly Charged Ring Q Method 1:Divide into point charges and add up contributions due to each charge

  11. Potential of a Uniformly Charged Ring Q Method 2:Integrate electric field along a path

  12. Potential of a Uniformly Charged Ring Q What is V for z>>R ? The same as for a point charge!

  13. Potential of a Uniformly Charged Disk one ring: integrate:

  14. Potential of a Uniformly Charged Disk Can find :

  15. Potential Difference in an Insulator A B 1 2 3 4 5 Electric field in capacitor filled with insulator:Enet=Eplates+Edipoles Eplates=const (in capacitor) Edipolescomplicated f(x,y,z) Edipoles,A Edipoles,B Travel from B to A: Edipolesis sometimes parallel to dl, and sometimes antiparallel to dl

  16. Potential Difference in an Insulator Instead of traveling through inside – travel outside from B to A: Edipoles, average A B Enet=Eplates+Edipoles,average Enet< Eplates Effect of dielectric is to reduce the potential difference.

  17. Dielectric Constant Electric field in capacitor filled with insulator:Enet=Eplates-Edipoles K – dielectric constant

  18. Dielectric Constant Inside an insulator: Dielectric constant for various insulators: vacuum 1 (by definition) air 1.0006 typical plastic 5 NaCl 6.1 water 80 strontium titanate 310

  19. Potential Difference in a Capacitor with Insulator s

  20. Potential Difference in Partially Filled Capacitor d K -Q +Q s A B x

  21. Energy Density of Electric Field Energy can be stored in electric fields (for small s) volume E Field energy density: (J/m3) Energy expended by us was converted into energy stored in the electric field

  22. Energy Density of Electric Field In the previous slide, the “system” is the set of two plates. Work, Wexternal > 0, is done on the system by you – part of the “surroundings.” If the force exerted by you just offsets the attractive force, Fby-plates, so that the plate moves with no gain in KE,

  23. Electric Field and Potential 90V 100V x Ex 1 mm

  24. Exercise Suppose in some area of space V(x,y,z)=x2+yz. What is E(x,y,z)?

  25. Potential Inside a Uniformly Charged Hollow Sphere =0 In general, integration path may be complex

  26. Electron-Volt (eV) – Unit of Energy What is the change in electric potential energy associated with moving an electron from 1Å to 2Å from a proton? If an electron moves through a potential difference of 1 V there is a change in electric potential energy of 1 eV. 1 eV = e.(1 V) = (1.6.10-19 C)(1 V) = 1.610-19J

  27. Shifting the Zero Potential In most cases we are interested in V, not the absolute values of V