Chapter 24

1. Chapter 24 Gauss’s Law

2. Intro • Gauss’s Law is an alternative method for determining electric fields. • While it stem’s from Coulomb’s law, Gauss’s law is more convenient for highly symmetric charge distributions and allows qualitative reasoning to simply complicated scenarios.

3. 24.1 Electric Flux • Ch 23 dealt with electric field lines qualitatively. • Consider a uniform electric field passing through a surface of area A • Remember that the number of lines passing through a unit area is proportional to the magnitude of the E-field.

4. 24.1 • We can then draw the conclusion that the total number of electric field lines penetrating the surface is proportional to the product EA • The product of E and perpendicular A will be called Electric Flux • Electric Flux (N.m2/C) is proportional to the number of electric field lines penetrating some surface.

5. 24.1 • Example 24.1 • If the surface is not perpendicular to the E-field, it will have less flux than ΦE = EA • As the area is rotated away from perpendicular, fewer lines will pass through it.

6. 24.1 • The electric flux through a given area will be the dot product of the E and A • A is a vector? • vector A points in a direction normal to the surface area. (See Diagram)

7. 24.1 • This all assumes a uniform field, although in more general situations, the field can vary over a given surface. • Our definition of flux now only applies to small elements of the surface area (ΔA) • The electric flux through each element is therefore

8. 24.1 • The total flux through the entire surface will be the sum of flux through each element as the elements become infinitely small. • A surface integral, evaluated over the surface in question. Generally depends on the field pattern and the surface.

9. 24.1 • Often we are interested in evaluating flux through a closed surface.

10. 24.1 • Elements where the electric field lines enter the surface experience negative flux (90o<θ<180o) • Elements where the electric field skims the surface experience zero flux (θ = 90o) • Elements where the electric field leaves the surface experience positive flux (θ < 90o)

11. 24.1 • The net flux on a closed surface is equal to the number of lines leaving the surface minus the number of lines entering the surface. • More lines leaving, net ΦE is positive • More lines entering, net ΦE is negative • Net Flux-

12. 24.1 • The symbol indicates integration over a closed surface. • En is the component of the E field that is normal to the surface. • If the entire field is normal to each point of the surface, and constant in magnitude the calculation is very straightforward as in Ex. 24.1

13. 24.1 • Quick quizzes p. 742 • Example 24.2

14. 24.2 Gauss’s Law • Gauss’s Law gives the general relationship between the net electric flux through a closed (Gaussian) surface and the charge enclosed by the surface. • Again we will look at a positive charge q at the center of a sphere of radius r.

15. 24.2 • We found that flux • Simplifying Since

16. 24.2 • We can use to verify our results from example 24.1 • Consider multiple Gaussian surfaces • The flux through each of the surfaces should be the same.

17. 24.2 • The net flux through any closed surface surrounding a point charge is given by q/εo and is independent to the shape of that surface.

18. 24.2 • Now consider a point charge located outside the closed arbitrary surface. • Every field line that enters the surface also leaves the surface. • The net electric flux through a closed surface that surrounds no charge is zero.

19. 24.2 • Gauss’s Law- The Net Electric Flux through any closed surface is • While qin represents just the enclosed charges E represents the TOTAL E-field through an element of area, from both inside charge and outside charge.

20. 24.2 • Gauss’s Law can be used to solve for E, for a system of charges or a continous distribution of charge. • It is however limited to situations of high symmetry. By choosing the shape of the gaussian surface carefully, the integral in the equation becomes very simple. • Quick Quizzes p. 745 • Example 24.3

21. 24.3 Applications of Gauss’s Law • Applied to charge distributions of high symmetry. • A gaussian surface is not a real surface, but rather one that is chosen that will simplify the integral by- • Making the value of E constant over the entire surface • The dot product has no angle because E and dA are parallel

22. 24.3 • The dot product is zero because E and dA are perpendicular. • The field can be argued to be zero over the entire surface. • Examples 24.4-24.8

23. 24.4 Conductors In Electrostatic Equilibrium • A conductor has charges (e-) that are unbound to atoms/molecules, therefore free to move. • Electrostatic Equilibrium- there is no net motion of the charge within the conductor • A conductor in equilibrium has the following four properties.

24. 24.4 • 1) The electric field is zero everywhere inside the conductor. • Consider a conducting slab. • If there is a field inside the conductor, charges present would experience an electric force and therefore acceleration, which would not be considered equilibrium

25. 24.4 • 2) If an isolated conductor carries a charge, the charge resides on its surface. • Charged Conductor • No E-Field inside • Gaussian Surface as near as possible to actual surface, qin = 0 • The charge must reside on the surface.

26. 24.4 • The electric field just outside a charged conductor is perpendicular to the surface and has magnitude σ/εo where σis the surface charge density at that point. • If there is any component of E that is parallel to the surface of the object, free electrons would accelerate, indicating no equilibrium

27. 24.4 • We can Determine the E field just outside the surface by choosing the cylindrical gaussian surface whose two flat sides are just inside/outside of the charged surface. • Since the E-field inside is zero from property #1

28. 24.4

29. 24.4 • 4) On an irregularly shaped conductor, the surface charge density is greatest at locations where the radius of curvature is smallest. • The 4th property will be verified in Ch 25. Quick quiz 24.6 Example 24.10