Karl Friedrich Gauss (1777-1855) – German mathematician

1. Ch 24 – Gauss’s Law Karl Friedrich Gauss (1777-1855) – German mathematician

2. Ch 24 – Gauss’s Law Already can calculate the E-field of an arbitrary charge distribution using Coulomb’s Law. Gauss’s Law allows the same thing, but much more easily… … so long as the charge distribution is highly symmetrical. Karl Friedrich Gauss (1777-1855) – German mathematician

3. Ch 24 – Gauss’s Law For example, in Ch 23: We found the E-fields in the vicinity of continuous charge distributions by integration… booooo: E-field of charged disc (R>>x): Now, we’ll learn an easier way.

4. Ch 24.1 – Electric Flux • Sounds fancy, but it’s not hard • Electric Fluxmeasures how much an electric field wants to “push through” or “flow through” some arbitrary surface area • We care about flux because it makes certain calculations easier.

5. Ch 24.1 – Electric Flux – Case 1 Easiest case: • The E-field is uniform • The plane is perpendicular to the field Electric Flux

6. Ch 24.1 – Electric Flux – Case 1 Easiest case: • The E-field is uniform • The plane is perpendicular to the field Electric Flux Flux depends on how strong the E-field is and how big the area is.

7. Ch 24.1 – Electric Flux – Case 2 Junior Varsity case: • The field is uniform • The plane is not perpendicular to the field

8. Ch 24.1 – Electric Flux – Case 2 Junior Varsity case: • The field is uniform • The plane is not perpendicular to the field Flux depends on how strong the E-field is, how big the area is, and the orientation of the area with respect to the field’s direction.

9. Ch 24.1 – Electric Flux – Case 2 And, we can write this better using the definition of the “dot” product. where:

10. Ch 24.1 – Electric Flux – Case 2 Quick Quiz: What would happen to the E-flux if we change the orientation of the plane?

11. Ch 24.1 – Electric Flux – Case 3 Varsity (most general) case: • The E-field is not uniform • The surface is curvy and is not perpendicular to the field

12. Ch 24.1 – Electric Flux – Case 3 Imagine the surface A is a mosaic of little tiny surfaces ΔA. Pretend that each little ΔA is so small that it is essentially flat.

13. Ch 24.1 – Electric Flux – Case 3 Then, the flux through each little ΔA is just: is a special vector. It points in the normal direction and has a magnitude that tells us the area of ΔA .

14. Ch 24.1 – Electric Flux – Case 3 So… to get the flux through the entire surface A, we just have to add up the contributions from each of the little ΔA’s that compose A.

15. Ch 24.1 – Electric Flux – Case 3 Electric Flux through an arbitrary surface caused by a spatially varying E-field.

16. Ch 24.1 – Electric Flux – Flux through a Closed Surface • The vectors dAi point in different directions • At each point, they are perpendicular to the surface • By convention, they point outward

17. Electric Flux: General Definition

18. Electric Flux: General Definition • E-Flux through a surface depends on three things: • How strong the E-field is at each infinitesimal area. • How big the overall area A is after integration. • The orientation between the E-field and each infinitesimal area.

19. Electric Flux: General Definition Flux can be negative, positive or zero! -The sign of the flux depends on the convention you assign. It’s up to you, but once you choose, stick with it.

20. Quick Quiz: which little area experiences the most flux?

21. Quick Quiz: what are the three things on which E-flux depends?

22. Ch 24.1 – Electric Flux – Calculating E-Flux • The surface integral means the integral must be evaluated over the surface in question… more in a moment. • The value of the flux will depend both on the field pattern and on the surface • The units of electric flux are N.m2/C

23. Ch 24.1 – Electric Flux • The net electric flux through a surface is directly proportional to the number of electric field lines passing through the surface.

24. EG 24.1 – Flux through a Cube Assume a uniform E-field pointing only in +x direction Find the net electric flux through the surface of a cube of edge-length l, as shown in the diagram.

25. Ch 24.2 – Gauss’s Law Gauss’s Law is just a flux calculation We’re going to build imaginary surfaces – called Gaussian surfaces – and calculate the E-flux. Gauss’s Law only applies to closed surfaces. Gauss’s Law directly relates electric flux to the charge distribution that creates it.

26. Ch 24.2 – Gauss’s Law Gauss’s Law

27. Ch 24.2 – Gauss’s Law Gauss’s Law The net E-flux through a closed surface Charge inside the surface

28. Ch 24.2 – Gauss’s Law • In other words… • 1. Draw a closed surface around a some charge. • Set up Gauss’s Law for the surface you’ve drawn. • Use Gauss’s Law to find the E-field. You get to choose the surface – it’s a purely imaginary thing.

29. Quick Quiz Which surface – S1, S2 or S3 – experiences the most electric flux?

30. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law Let’s calculate the net flux through a Gaussian surface. Assume a single positive point charge of magnitude q sits at the center of our imaginary Gaussian surface, which we choose to be a sphere of radius r.

31. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law At every point on the sphere’s surface, the electric field from the charge points normal to the sphere… why? This helps make our calculation easy.

32. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law At every point on the sphere’s surface, the electric field from the charge points normal to the sphere… why? This helps make our calculation easy.

33. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law Now we have: But, because of our choice for the Gaussian surface, symmetry works in our favor. The electric field due to the point charge is constant all over the sphere’s surface. So…

34. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law This, we can work with. We know how to find the magnitude of the electric field at the sphere’s surface. Just use Coulomb’s law to calculate the E-field due to a point charge a distance r away from the charge.

35. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law Thus:

36. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law Thus: And, this surface integral is easy.

37. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law Therefore: But, we can rewrite Coulomb’s constant.

38. Ch 24.2 – Gauss’s Law – confirming Gauss’s Law Therefore: But, we can rewrite Coulomb’s constant. Thus, we have confirmed Gauss’s law:

39. A few more questions • If the electric field is zero for all points on the surface, is the electric flux through the surface zero? • If the electric flux is zero for a closed surface, can there be charges inside the surface? • What is the flux through the surface shown? Why?

40. EG 24.2 – Flux due to a Point Charge • A spherical surface surrounds a point charge. • What happens to the total flux through the surface if: • the charge is tripled, • the radius of the sphere is doubled, • the surface is changed to a cube, and • the charge moves to another location inside the surface?

41. Ch 24.3 – Applying Gauss’s Law Gauss’s Law can be used to (1) find the E-field at some position relative to a known charge distribution, or (2) to find the charge distribution caused by a known E-field. In either case, you must choose a Gaussian surface to use.

42. Ch 24.3 – Applying Gauss’s Law • Choose a surface such that… • Symmetry helps: the E-field is constant over the surface (or some part of the surface) • The E-field is zero over the surface (or some portion of the surface) • The dot product reduces to EdA (the E-field and the dA vectors are parallel) • 4. The dot product reduces to zero (the E-field and the dA vectors are perpendicular)

43. EG 24.3 – Spherical Charge Distribution • An insulating solid sphere of radius a has a uniform volume charge density ρ and carries total charge Q. • Find the magnitude of the E-field at a point outside the sphere • Find the magnitude of the E-field at a point inside the sphere

44. EG 24.3 – Spherical Charge Distribution

45. EG 24.4 – Spherical Charge Distribution Find the E-field a distance r from a line of positive charge of infinite length and constant charge per unit length λ.

46. EG 24.5 – Spherical Charge Distribution Find the E-field due to an infinite plane of positive charge with uniform surface charge density σ

47. Ch 24.4 – Conductors in Electrostatic Equilibrium • In an insulator, excess charge stays put. • Conductors have free electrons and, correspondingly, have different electrostatic characteristics. • You will learn four critical characteristics of a conductor in electrostatic equilibrium. • Electrostatic Equilibrium – no net motion of charge.

48. Ch 24.4 – Conductors in Electrostatic Equilibrium • Most conductors, on their own, are in electrostatic equilibrium. • That is, in a piece of metal sitting by itself, there is no “current.”

49. Ch 24.4 – Conductors in Electrostatic Equilibrium Four key characteristics • The E-field is zero at all points inside a conductor, whether hollow or solid. • If an isolated conductor carries excess charge, the excess charge resides on its surface. • The E-field just outside a charged conductor is perpendicular to the surface and has magnitudeσ/ε0, where σ is the surface charge density at that point. • Surface charge density is biggest where the conductor is most pointy.

50. Ch 24.4 – Conductors (cont.) – Justifications Einside = 0 • Place a conducting slab in an external field, E. • If the field inside the conductor were not zero, free electrons in the conductor would experience an electrical force. • These electrons would accelerate. • These electrons would not be in equilibrium. • Therefore, there cannot be a field inside the conductor.