Chapter 22: Gauss' Law

What are three cases in which there is zero net charge inside a box and no net electric flux through the surface of the box?

1) An empty box with E = 0. 2) A box containing one positive and one equal magnitude negative point charge. 3) An empty box immersed in a uniform electric field (so no charge on the inside, but charge on the outside)

What are the three qualitative statements of Gauss' Law?

1. Whether there is a net outward or inward electric flux through a closed surface depends on the sign of the enclosed charge. 2. Charges outside the surface do not give a net electric flux through the surface. 3. The net electric flux is directly proportional to the net amount of charge enclosed within the surface but is otherwise independent of the size of the closed surface.

What is electric flux?

Flux is a measure of the number of field lines passing through an area.

For what distribution of charges and closed surfaces is Gauss' law valid for?

Gauss' law is valid for any distribution of charges and for any closed surface.

What does Gauss' law state?

Gauss' law states that the total electric flux through any closed surface (a surface enclosing a definite volume) is proportional to the total (net) electric charge inside the surface. The total electric flux through a closed surface is equal to the total (net) electric charge inside the surface, divided by ε0 (epsilon naught).

Does the net electric flux due to a single point charge inside the box depend on the size of the box? Why are or why not?

No it doesn't. Although the magnitude of E in a box twice as large would decrease by a factor of 1/4 due to 1/r^2, the surface area through which E travels increases by a factor of 4, thereby cancelling out the discrepancy.

For a point charge inside a spherical surface, does the flux depend on the radius of the sphere?

No, it only depends on the charge q enclosed by the sphere.

When can electric field lines begin or end inside a region of space?

Only when there is charge in that region.

How does the charge inside a box relate to the direction of electric flux through the box's surface?

Positive charge inside the box goes with an outward electric flux through the box's surface, and negative charge inside goes with an inward electric flux. If there is zero charge inside the box and E = 0 everywhere, there is no electric flux into or out of the box.

What does the A vector represent when talking about fluid flow through a surface? In which direction does it point?

The A vector represents the the area of the plane, of the vector area A. Its magnitude is the area of the surface. It is always pointing outwards perpendicular to the surface of A, which corresponds to a positive value of flux.

What happens when excess charge is placed on a conductor and is at rest?

The charge resides entirely on the surface, not in the interior of the material. (By excess we mean charges other than the ions and free electrons that make up the neutral conductor.) In an electrostatic situation (with all charges at rest) the electric field E at every point in the interior of a conducting material is zero. If E were not zero, the excess charge would move.

What is the relationship between the net electric flux through the surface of the box and the magnitude of the net charge enclosed by the box?

They are directly proportional.

What's the difference between Gauss' law and Coulomb's law?

They are equivalent; Gauss' law provides a different way to express the relationship between electric charge and electric field.