1. Dr. Jie ZouPHY 13611 Chapter 24 Gauss’s Law (cont.)

2. Dr. Jie ZouPHY 13612 Outline Gauss’s law (24.2) Application of Gauss’s law to various charge distributions (24.3)

3. Dr. Jie ZouPHY 13613 Gauss’s law Gauss’s law describes a general relationship between the net electric flux through a closed surface and the charge enclosed by the surface. Closed surface: often called a gaussian surface.

4. Dr. Jie ZouPHY 13614 Let’s begin with one example. A spherical gaussian surface of radius r surrounding a point charge q. The magnitude of the electric field everywhere on the surface of the sphere is E = k e q/r 2. The electric field is  to the surface at every point on the surface. Net electric flux through such gaussian surface is

5. Dr. Jie ZouPHY 13615 A non-spherical closed surface surrounding a point charge As we discussed in the previous section, the electric flux is proportional to the number of electric field lines passing through a surface. The number of lines through S 1 is equal to the number of lines through the nonspherical surfaces S 2 and S 3. The net flux through any closed surface surrounding a point charge q is given by q/  0 and is independent of the shape of that surface.

6. Dr. Jie ZouPHY 13616 A point charge located outside a closed surface Any electric field line that enters the surface leaves the surface at another point. The net electric flux through a closed surface that surrounds no charge is zero. Revisit Example 24.2

7. Dr. Jie ZouPHY 13617 Example: Problem #14, P. 762 Calculate the total electric flux through the paraboloidal surface due to a constant electric field of magnitude E 0 in the direction shown in the figure.

8. Dr. Jie ZouPHY 13618 Now let’s consider a more general case. The net electric flux through S is  E = q 1 /  0. The net electric flux through S’ is  E = (q 2 + q 3 )/  0. The net electric flux through S” is  E = 0. Charge q 4 does not contribute to the flux through any surface because it is outside all surfaces.

9. Dr. Jie ZouPHY 13619 Gauss’s law Gauss’s law: the net flux through any closed surface is q in = the net charge inside the gaussian surface. E = the (total) electric field at any point on the surface, which includes contributions from charges both inside and outside the surface. Pitfall prevention: Zero flux is not zero field.

10. Dr. Jie ZouPHY 136110 Application of Gauss’s law to various charge distributions Example 24.5 A spherically symmetric charge distribution: An insulating solid sphere of radius a has a uniform volume charge density  and carries a total positive charge Q. (A) Calculate the magnitude of the electric field at a point outside the sphere. (B) Find the magnitude of the electric field at a point inside the sphere. Answer: (A) E = k e Q/r 2, r > a; (B), r < a.

11. Dr. Jie ZouPHY 136111 Application of Gauss’s law to various charge distributions Example 24.7 A cylindrical symmetric charge distribution: Find the electric field a distance r from a line of positive charge of infinite length and constant charge per unit length. Answer:

12. Dr. Jie ZouPHY 136112 Homework Ch. 24, P. 762, Problems: #12, 14, 29.