Gauss’s Law

1. Gauss’s Law What does it mean? How do we use it? Dominic Berry University of Waterloo Griffith University 8 February, 2011

2. What does it mean? • First we need the concept of flux. Area A

3. Electric field Area A What does it mean? • First we need the concept of flux.

4. What does it mean? • First we need the concept of flux. Electric field Area A Flux is just electric field times area

5. What does it mean? • First we need the concept of flux. If electric field does not pass through the area, flux is zero.

6. What does it mean? • First we need the concept of flux. In general we use a normal vector to the plane, .

7. What does it mean? • First we need the concept of flux. For more general angles the flux varies as cos.

8. What does it mean? • First we need the concept of flux. For more general angles the flux varies as cos.

9. What does it mean? • The total flux through a closed surface.

10. What does it mean? • The total flux through a closed surface. • We use the convention that the normal always points outward.

11. What does it mean? • The total flux through a closed surface. • We use the convention that the normal always points outward.

12. What does it mean? • The total flux through a closed surface. • We use the convention that the normal always points outward. • For the four sides,

13. What does it mean? • The total flux through a closed surface. • We use the convention that the normal always points outward. • For the four sides, • For the top,

14. What does it mean? • The total flux through a closed surface. • We use the convention that the normal always points outward. • For the four sides, • For the top, • For the bottom,

15. What does it mean? • The total flux through a closed surface. • We use the convention that the normal always points outward. • For the four sides, • For the top, • For the bottom, • The total flux is

16. What does it mean? • What does the integral mean? • The circle indicates an integral over the closed surface.

17. What does it mean? • What does the integral mean? • The circle indicates an integral over the closed surface. • In practice we will not have to evaluate the interval.

18. What does it mean? • What does the integral mean? • The circle indicates an integral over the closed surface. • In practice we will not have to evaluate the interval. • We break the surface up into sections where the flux is easy to calculate.

19. What does it mean? • What does the integral mean? • The circle indicates an integral over the closed surface. • In practice we will not have to evaluate the interval. • We break the surface up into sections where the flux is easy to calculate. In principle sum over infinitesimal elements .

20. What does it mean? • The full Gauss’s law. • The left side is the total flux out through the surface.

21. What does it mean? • The full Gauss’s law. • The left side is the total flux out through the surface. • The right side is proportional to the charge, q, inside the surface. +q

22. What does it mean? • The full Gauss’s law. • The left side is the total flux out through the surface. • The right side is proportional to the charge, q, inside the surface. • The constant, 0, is the usual vacuum permittivity. +q

23. How do we use it? • For example, consider a charge +q. +q r

24. How do we use it? • For example, consider a charge +q. • We choose a spherical surface. +q r

25. How do we use it? • For example, consider a charge +q. • We choose a spherical surface. • By spherical symmetry the electric field must be directed radially outwards. +q r

26. How do we use it? • For example, consider a charge +q. • We choose a spherical surface. • By spherical symmetry the electric field must be directed radially outwards. • The magnitude of the electric field must be constant on the surface. +q r

27. How do we use it? • For example, consider a charge +q. • We choose a spherical surface. • By spherical symmetry the electric field must be directed radially outwards. • The magnitude of the electric field must be constant on the surface. • The flux is just EA. +q r

28. How do we use it? • For example, consider a charge +q. • We choose a spherical surface. • By spherical symmetry the electric field must be directed radially outwards. • The magnitude of the electric field must be constant on the surface. • The flux is just EA. • Gauss’s law gives +q r

29. How do we use it? • For example, consider a charge +q. • We choose a spherical surface. • By spherical symmetry the electric field must be directed radially outwards. • The magnitude of the electric field must be constant on the surface. • The flux is just EA. • Gauss’s law gives +q r

30. How do we use it? • For example, consider a charge +q. • We choose a spherical surface. • By spherical symmetry the electric field must be directed radially outwards. • The magnitude of the electric field must be constant on the surface. • The flux is just EA. • Gauss’s law gives +q r

31. How do we use it? • Consider a shell of charge +q. • We choose a spherical surface. • By spherical symmetry the electric field must be directed radially outwards. • The magnitude of the electric field must be constant on the surface. • The flux is just EA. • Gauss’s law gives +q r

32. How do we use it? • Consider a shell of charge +q. • We choose a spherical surface. • By spherical symmetry the electric field must be directed radially outwards. • The magnitude of the electric field must be constant on the surface. • The flux is just EA. • Gauss’s law gives +q r

33. How do we use it? General procedure: • Choose a surface corresponding to the symmetry of the problem. • Break the surface up into subsurfaces where the electric field is either • constant and parallel to the normal, or • perpendicular to the normal. • Evaluate the total flux using the electric field as a free parameter. • Solve Gauss’s law for E. +q r http://www.dominicberry.org/presentations/gauss.ppt