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2. Chapter 23 Gauss’ Law

3. Main Points of Chapter 23 Electric flux Gauss’ law Using Gauss’ law to determine electric fields Conductors and electric fields Testing Gauss’ and Coulomb’s laws

4. 23-1 What Does Gauss’ Law Do? Imagine field lines emanating from a positive charge. Now imagine a sphere of tissue paper around the charge. How many field lines penetrate the tissue? It doesn’t really matter how many we draw in the first place, as long as we are consistent; they all go through. Now imagine the charge being off- center; all the lines still go through:

5. 23-1 What Does Gauss’ Law Do? Suppose the tissue is some shape other than spherical, but still surrounds the charge. All the field lines still go through: Now, imagine the paper is crinkled and overlaps itself; how shall we deal with the lines that pierce the tissue three times? Notice that they go out twice and in once – if we subtract the “ins” from the “outs” we are left with one line going out, which is consistent with the other situations.

6. 23-1 What Does Gauss’ Law Do? Now, look at an open (flat) sheet. If it is perpendicular to the field, the maximum number of lines penetrates: If it is at an angle, fewer lines penetrate:

7. 23-1 What Does Gauss’ Law Do? The number of field lines piercing the surface is proportional to the surface area, the orientation, and the field strength. If we stop counting lines and just use the field strength itself, we can define the electric flux through an infinitesimal area: Integrating gives the total flux: (23-1)

8. 23-1 What Does Gauss’ Law Do? (23-3) For a closed surface, we can uniquely define the direction of the normal to the surface as pointing outwards and define: Then: Note the circle on the integral sign, which means that the integration is over a closed surface.

9. 23-1 What Does Gauss’ Law Do? Note that the surface does not have to be made of real matter – it is a surface that we can imagine, but that does not have to exist in reality. This kind of imaginary surface is called a Gaussian surface. We can imagine it to be any shape we want; it is very useful to choose one that makes the problem you are trying to solve as easy as possible.

10. 23-2 Gauss’ Law The electric flux through a closed surface that encloses no net charge is zero. Without further ado, we can state Gauss’ law:

11. 23-2 Gauss’ Law The electric flux through a Gaussian sphere with a single point charge at the center is easily calculated, as the field is known: (23-4)

12. 23-2 Gauss’ Law But the result would be the same if the surface was not spherical, or if the charge was anywhere inside it! Therefore, we can quickly generalize this to any surface and any charge distribution; all can be considered as a collection of point or infinitesimal charges: (23-5) Here, Q is the total net charge enclosed by the surface.

13. 23-3 Using Gauss’ Law to Determine Electric Fields Problem-solving techniques: 1. Make a sketch. 2. Identify any symmetries. 3. Choose a Gaussian surface that matches the symmetry – that is, the electric field is either parallel to the surface or constant and perpendicular to it. 4. The correct choice in 3 should allow you to get the field outside the integral. Then solve.

14. 23-4 Conductors and Electric Fields In conductors, the charges are free to move if there is an external electric field exerting a force on them. Therefore, in equilibrium, there is no static field inside a conductor. This also means that the external electric field is perpendicular to the conductor at its surface.

15. 23-4 Conductors and Electric Fields What if the conductor is charged – where does the excess charge go? By making a Gaussian surface very close to, but just under, the surface of the conductor, we see that any excess charge must lie on the outside of the conductor.

16. 23-4 Conductors and Electric Fields What if there is a cavity inside the conductor, and that cavity has charges in it? The field inside the conductor must still be zero, so charges will be induced on the inner surface of the cavity and the outer surface of the conductor:

17. 23-4 Conductors and Electric Fields Electrostatic Fields Near Conductors Looking at the electrostatic field very near a conductor, we find: and therefore: (23-12) The electric field is perpendicular to the surface, and where the charge density is higher, the field is larger.

18. 23-4 Conductors and Electric Fields 1. The electrostatic field inside a conductor is zero. 2. The electrostatic field immediately outside a conductor is perpendicular to the surface and has the value σ/ε 0 where σ is the local surface charge density. 3. A conductor in electrostatic equilibrium—even one that contains nonconducting cavities—can have charge only on its outer surface, as long as the cavities contain no net charge. If there is a net charge within the cavity, then an equal and opposite charge will be distributed on the surface of the conductor that surrounds the cavity. To summarize:

19. 23-5 Are Gauss’ and Coulomb’s Laws Correct? An experiment to validate Gauss’ law (that there is no charge within a conductor) can be done as follows: Need a hollow conducing sphere, a small conducting ball on an insulating rod, and an electroscope attached to the surface of the conductor.

20. 23-5 Are Gauss’ and Coulomb’s Laws Correct? Charge the small sphere and hold it inside the shell without touching. Induced charge will be on outside of shell and on electroscope. Now touch the inside of the shell with the small sphere. Charge will flow onto it until it is neutral, leaving the shell with a net positive charge.

21. 23-5 Are Gauss’ and Coulomb’s Laws Correct? Finally, remove the rod. The electroscope leaves do not move, indicating that the excess charge resides on the outside of the shell.

22. 23-5 Are Gauss’ and Coulomb’s Laws Correct? This table shows the results of such experiments looking for a deviation from an inverse-square law:

23. 23-5 Are Gauss’ and Coulomb’s Laws Correct? One problem with the above experiments is that they have all been done at short range, 1 meter or so. Other experiments, more sensitive to cosmic-scale distances, have been done, testing whether Coulomb’s law has the form: No evidence for a nonzero μ has been found.

24. Summary of Chapter 23 Electric flux due to field intersecting a surface S: (23-1) Gauss’ law relates flux through a closed surface to charge enclosed: (23-5) Can use Gauss’ law to find electric field in situations with a high degree of symmetry

25. Summary of Chapter 23, cont. Properties of conductors: 1. Electric field is zero inside 2. Field just outside conductor is perpendicular to surface 3. Excess charge resides on the outside of a conductor, unless there is a nonconducting cavity in it; in that case, there is an induced charge on both surfaces Gauss’ law has been verified to a very high degree of accuracy