Gauss’ Law

1. Gauss’ Law

2. Class Objectives • Introduce the idea of the Gauss’ law as another method to calculate the electric field. • Understand that the previous method of calculating the electric field strength does not consider symmetry. • Consider the different types of symmetry.

3. Class Objectives • Introduce the idea of the Gaussian surface. • Define the properties of Gaussian surfaces. • Show how to choose a Gaussian surface. • Show how Gaussian surfaces can be used to take advantage of symmetry.

4. Class Objectives • Show that the results of Gauss’s law are the same as the standard results but quicker and easier!

5. Student Objectives • Be able to use Gauss’ law to calculate the electric field for various objects. • Be able to determine the type of symmetry in the problem and hence the type of Gaussian surface to be used.

6. Gauss’ Law • Symmetry in problems arise naturally.

7. Gauss’ Law • Symmetry in problems arise naturally. • Gauss’ law is an alternate method to coulomb’s law where there is symmetry.

8. Gauss’ Law • Outline of Gauss’ Law • Central idea is a hypothetical closed surface called a Gaussian surface. • The surface can be chosen any shape to complement the symmetry. Eg cylinder or sphere. • Gauss’ law relates the electric field at a point on the closed surface to the net charge enclosed by the surface.

9. Gauss’ Law • The amount of charge enclosed is measured by the amount of flux passing through the surface.

10. Gauss’ Law • Example of a spherical Gaussian closed surface.

11. Electric flux • The electric flux is a measure of the number of field lines passing an area. • The flux is represented by the symbol .

12. Electric flux • The electric flux is a measure of the number of field lines passing an area. • The flux is represented by the symbol . • Assuming an arbitrary Gaussian surface in a field.

13. Electric flux • If we divide the surface into small squares of area then the curvature of the surface can be ignored if the squares are taken small enough.

15. Electric flux • If we divide the surface into small squares of area then the curvature of the surface can be ignored if the squares are taken small enough. • By convention we assign an area vector . • The direction of which is perpendicular to the surface of the element directed away from its interior.

16. Electric flux • Because can be made arbitrarily small, may be taken as constant for a given square. • The flux for a Gaussian surface is given as

17. Electric flux • Because can be made arbitrarily small, may be taken as constant for a given square. • The flux for a Gaussian surface is given as • The circle indicates that the integral is to be taken over the entire closed surface.

18. Electric flux • The follow diagrams gives three cases for the relative orientation of the field and dA. The angle measured is the smallest angle between the vectors.

19. Electric flux • Note that all of the elements on the surface do not contribution to the flux.

20. Electric flux • Note that all of the elements on the surface do not contribution to the flux.

21. Gauss’ Law • Guass’ Law is often written as • Where is the sum of the enclosed charges.

22. Gauss’ Law Choosing a Gaussian surface

23. Gauss’ Law • Choosing a Gaussian surface is not hard but subtle.

24. Gauss’ Law • Choosing a Gaussian surface is not hard but subtle. • A simple surface must be chosen to take advantage of symmetry. • Consider the following rules as a guide.

25. Gauss’ Law • Choose the surface perpendicular to the field so the E and dA are parallel.

26. Gauss’ Law • Choose the surface perpendicular to the field so the E and dA are parallel. • Choose so that points on the surface are equal distance away from the charge so the E doesn’t vary.

27. Gauss’ Law • Choose the surface perpendicular to the field so the E and dA are parallel. • Choose so that points on the surface are equal distance away from the charge so the E doesn’t vary. • If this is not possible (1 and 2) choose a surface such that the dot product is zero. Ie. They are perpendicular.

28. Gauss’ Law and Coulomb’s Law • Consider a point charge q.

29. Gauss’ Law and Coulomb’s Law • Consider a point charge q. • For symmetry we use a spherical Gaussian surface.

30. Gauss’ Law and Coulomb’s Law • the chosen Gaussian surface holds true for the first two guidelines.

31. Gauss’ Law and Coulomb’s Law dA • the chosen Gaussian surface holds true for the first two guidelines. • E and dA are parallel. • Points on the surface are equal distance away from the charge so the E doesn’t vary.

32. Gauss’ Law • From Gauss’ Law

33. Gauss’ Law • From Gauss’ Law • Since the two vectors are parallel.

34. Gauss’ Law • From Gauss’ Law • Since the two vectors are parallel. • Also we note that the enclosed charge is simply q. So we can write that,

35. Gauss’ Law • The area of the closed surface is the area of a circle. So that

36. Gauss’ Law • The area of the closed surface is the area of a circle. So that • Therefore

37. Conductor in an Electric Field

38. Conductor in an Electric Field • For the cases to investigated, we will consider the cases where the conductor is in equilibrium (electrostatics).

39. Conductor in an Electric Field • For the cases to investigated, we will consider the cases where the conductor is in equilibrium (electrostatics). • For a conductor in equilibrium we have following conditions:

40. Conductor in an Electric Field • The charge exists entirely on the surface of conductor(no charge is found within the body of the conductor).

41. Conductor in an Electric Field • The charge exists entirely on the surface of conductor(no charge is found within the body of the conductor). • The electric field within the conductor is zero.

42. Conductor in an Electric Field • The charge exists entirely on the surface of conductor(no charge is found within the body of the conductor). • The electric field within the conductor is zero. The charge distributes itself so as to get as far from each other as possible.

43. Conductor in an Electric Field • The external electric field is perpendicular to the surface of the conductor.

44. Conductor in an Electric Field • The external electric field is perpendicular to the surface of the conductor. If not it would cause the charges to move along the surface of the conductor.

45. Conductor in an Electric Field • The external electric field is perpendicular to the surface of the conductor. If not it would cause the charges to move along the surface of the conductor. • Note: unless the conductor is spherical, the charge does not distribute itself uniformly.

46. Conductor in an Electric Field • When a conductor is place in an external field, the mobile electrons experience a force pushing them in the opposite direction to the field E e

47. Conductor in an Electric Field • Thus making the top of the conductor positive and bottom negative. E + + + + + + + + + + e _ _ _ _ _ _ _ _ _ _ _

48. Conductor in an Electric Field • Thus making the top of the conductor positive and bottom negative. This sets up an internal electric which grows in strength as more electrons move. E + + + + + + + + + + Electric field caused by pd between the top and bottom of the conductor Eint e _ _ _ _ _ _ _ _ _ _ _

49. Conductor in an Electric Field • Electrons move until the two fields have the same magnitude. Hence the net electric field E=0. E + + + + + + + + + + Eint e _ _ _ _ _ _ _ _ _ _ _

50. Conductor in an Electric Field • Electrons move until the two fields have the same magnitude. Hence the net electric field E=0. The magnitude of the charge on the top & bottom of the conductor are the same. E E + + + + + + + + + + Eint e Eint _ _ _ _ _ _ _ _ _ _ _