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Flux and Gauss’s Law

Flux and Gauss’s Law. Spring 2008. Last Time: Definition – Sort of – Electric Field Lines. CHARGE. DIPOLE FIELD LINK. Field Lines  Electric Field. Last time we showed that. Ignore the Dashed Line … Remember last time .. the big plane?. s/2e 0. s/2e 0. s/2e 0. s/2e 0.

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Flux and Gauss’s Law

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  1. Flux andGauss’s Law Spring 2008

  2. Last Time: Definition – Sort of – Electric Field Lines CHARGE DIPOLE FIELD LINK

  3. Field Lines  Electric Field

  4. Last time we showed that

  5. Ignore the Dashed Line … Remember last time .. the big plane? s/2e0 s/2e0 s/2e0 s/2e0 s/2e0 s/2e0 E=0 E=s/e0 E=0 We will use this a lot!

  6. NEW RULES (Bill Maher) • Imagine a region of space where the ELECTRIC FIELD LINES HAVE BEEN DRAWN. • The electric field at a point in this region is TANGENT to the Electric Field lines that have been drawn. • If you construct a small rectangle normal to the field lines, the Electric Field is proportional to the number of field lines that cross the small area. • The DENSITY of the lines. • We won’t use this much

  7. What would you guess is inside the cube? • A positive charge • A negative charge • Can’t tell • An Alien • A car

  8. What about now? • A positive charge • A negative charge • Can’t tell • An Alien • A car

  9. How about this?? • Positive point charge • Negative point charge • Large Sheet of charge • No charge • You can’t tell from this

  10. Which box do you think contains more charge? • Top • Bottom • Can’t tell • Don’t care

  11. All of the E vectors in the bottom box are twice as large as those coming from the top box. The top box contains a charge Q. How much charge do you think is in the bottom box? • Q • 2Q • You can’t tell • Leave me alone.

  12. So far … • The electric field exiting a closed surface seems to be related to the charge inside. • But … what does “exiting a closed surface mean”? • How do we really talk about “the electric field exiting” a surface? • How do we define such a concept? • CAN we define such a concept?

  13. Mr. Gauss answered the question with.. Yup .. Gauss's Law

  14. Another QUESTION: Not Quite Solid Surface Given the electric field at EVERY point on a closed surface, can we determine the charges that caused it??

  15. A Question: • Given the magnitude and direction of the Electric Field at a point, can we determine the charge distribution that created the field? • Is it Unique? • Question … given the Electric Field at a number of points, can we determine the charge distribution that caused it? • How many points must we know??

  16. Still another question Given a small area, how can you describe both the area itself and its orientation with a single stroke!

  17. The “Area Vector” • Consider a small area. • It’s orientation can be described by a vector NORMAL to the surface. • We usually define the unit normal vector n. • If the area is FLAT, the area vector is given by An, where A is the area. • A is usually a differential area of a small part of a general surface that is small enough to be considered flat.

  18. E n En The “normal component” of the ELECTRIC FIELD

  19. E n En DEFINITION FLUX

  20. q We will be considering CLOSED surfaces The normal vector to a closed surface is DEFINED as positive if it points OUT of the surface. Remember this definition!

  21. “Element” of Flux of a vector E leaving a surface For a CLOSED surface: n is a unit OUTWARD pointing vector.

  22. q This flux is LEAVING the closed surface.

  23. Definition of TOTAL FLUX through a surface

  24. Flux is • A vector • A scaler • A triangle

  25. VisualizingFlux n is the OUTWARD pointing unit normal.

  26. Definition: A Gaussian Surface Any closed surface that is near some distribution of charge

  27. Remember Component of E perpendicular to surface. This is the flux passing through the surface and n is the OUTWARD pointing unit normal vector! n E q q A

  28. Flux is -EL2 ExampleCube in a UNIFORM Electric Field Flux is EL2 E area L Note sign E is parallel to four of the surfaces of the cube so the flux is zero across these because E is perpendicular to A and the dot product is zero. Total Flux leaving the cube is zero

  29. Simple Example r q

  30. Gauss’ Law Flux is total EXITING the Surface. n is the OUTWARD pointing unit normal. q is the total charge ENCLOSED by the Gaussian Surface.

  31. Simple ExampleUNIFORM FIELD LIKE BEFORE E No Enclosed Charge A A E E

  32. Q L Line of Charge

  33. Line of Charge From SYMMETRY E is Radial and Outward

  34. What is a Cylindrical Surface?? Ponder

  35. Drunk Looking at A Cylinder from its END Circular Rectangular

  36. Infinite Sheet of Charge +s h E cylinder We got this same result from that ugly integration!

  37. Materials • Conductors • Electrons are free to move. • In equilibrium, all charges are a rest. • If they are at rest, they aren’t moving! • If they aren’t moving, there is no net force on them. • If there is no net force on them, the electric field must be zero. • THE ELECTRIC FIELD INSIDE A CONDUCTOR IS ZERO!

  38. More on Conductors • Charge cannot reside in the volume of a conductor because it would repel other charges in the volume which would move and constitute a current. This is not allowed. • Charge can’t “fall out” of a conductor.

  39. Isolated Conductor Electric Field is ZERO in the interior of a conductor. Gauss’ law on surface shown Also says that the enclosed Charge must be ZERO. Again, all charge on a Conductor must reside on The SURFACE.

  40. Charged Conductors Charge Must reside on the SURFACE - - E=0 - - E - s Very SMALL Gaussian Surface

  41. Charged Isolated Conductor • The ELECTRIC FIELD is normal to the surface outside of the conductor. • The field is given by: • Inside of the isolated conductor, the Electric field is ZERO. • If the electric field had a component parallel to the surface, there would be a current flow!

  42. Isolated (Charged) Conductor with a HOLE in it. Because E=0 everywhere inside the conductor. So Q (total) =0 inside the hole Including the surface.

  43. A Spherical Conducting Shell withA Charge Inside. A Thinker!

  44. Insulators • In an insulator all of the charge is bound. • None of the charge can move. • We can therefore have charge anywhere in the volume and it can’t “flow” anywhere so it stays there. • You can therefore have a charge density inside an insulator. • You can also have an ELECTRIC FIELD in an insulator as well.

  45. E O r Example – A Spatial Distribution of charge. Uniform charge density = r = charge per unit volume (Vectors) A Solid SPHERE

  46. Outside The Charge R r O E Old Coulomb Law!

  47. Graph E r R

  48. s s ++++++++ ++++++++ A A E Charged Metal Plate E E is the same in magnitude EVERYWHERE. The direction is different on each side.

  49. Apply Gauss’ Law s s ++++++++ ++++++++ A A E Same result!

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