Chapter 23 Gauss’s Law Spring 2008

(23-1) In this chapter we will introduce the following new concepts: Chapter 23 Gauss’ law In this chapter we will introduce the following new concepts: The flux (symbol Φ ) of the electric field. Symmetry Gauss’ law We will then apply Gauss’ law and determine the electric field generated by: An infinite, uniformly charged insulating plane. An infinite, uniformly charged insulating rod. A uniformly charged spherical shell. A uniform spherical charge distribution. We will also apply Gauss’ law to determine the electric field inside and outside charged conductors. (23-1)

Last Time: Definition – Sort of – Electric Field Lines

Field Lines  Electric Field

Last time we showed that

Ignore the Dashed Line … Remember last time .. the big plane? s/2e0 s/2e0 s/2e0 s/2e0 s/2e0 s/2e0 E=0 E=s/e0 E=0 We will use this a lot!

NEW RULES Imagine a region of space where the ELECTRIC FIELD LINES HAVE BEEN DRAWN. The electric field at a point in this region is TANGENT to the Electric Field lines that have been drawn. If you construct a small rectangle normal to the field lines, the Electric Field is proportional to the number of field lines that cross the small area. The DENSITY of the lines. We won’t use this much

So far … The electric field exiting a closed surface seems to be related to the charge inside. But … what does “exiting a closed surface mean”? How do we really talk about “the electric field exiting” a surface? How do we define such a concept? CAN we define such a concept?

Mr. Gauss answered the question with.. Yup .. Gauss's Law

The “normal component” of the ELECTRIC FIELD

DEFINITION FLUX E n En

“Element” of Flux of a vector E leaving a surface For a CLOSED surface: n is a unit OUTWARD pointing vector.

Definition of TOTAL FLUX through a surface

Visualizing Flux n is the OUTWARD pointing unit normal.

Remember Component of E perpendicular to surface. This is the flux passing through the surface and n is the OUTWARD pointing unit normal vector! n E q q A

Example Cube in a UNIFORM Electric Field Flux is EL2 Flux is -EL2 area L Note sign E is parallel to four of the surfaces of the cube so the flux is zero across these because E is perpendicular to A and the dot product is zero. Total Flux leaving the cube is zero

Simple Example r q

Gauss’ Law n is the OUTWARD pointing unit normal. Flux is total EXITING the Surface. n is the OUTWARD pointing unit normal. q is the total charge ENCLOSED by the Gaussian Surface.

Line of Charge L Q

Materials Conductors THE ELECTRIC FIELD INSIDE A CONDUCTOR IS ZERO! Electrons are free to move. In equilibrium, all charges are a rest. If they are at rest, they aren’t moving! If they aren’t moving, there is no net force on them. If there is no net force on them, the electric field must be zero. THE ELECTRIC FIELD INSIDE A CONDUCTOR IS ZERO!

(23-6) e

Isolated Conductor Electric Field is ZERO in the interior of a conductor. Gauss’ law on surface shown Also says that the enclosed Charge must be ZERO. All charge on a Conductor must reside on The SURFACE.

Charged Conductors s E=0 E Charge Must reside on the SURFACE - - - - - Very SMALL Gaussian Surface

Charged Isolated Conductor The ELECTRIC FIELD is normal to the surface outside of the conductor. The field is given by: Inside of the isolated conductor, the Electric field is ZERO. If the electric field had a component parallel to the surface, there would be a current flow!!!

Isolated (Charged) Conductor with a HOLE in it. Because E=0 everywhere inside the conductor.

featureless sphere rotation axis observer Rotational symmetry Symmetry: We say that an object is symmetric under a particular mathematical operation (e.g. rotation, translation, …) if to an observer the object looks the same before and after the operation. Note: Symmetry is a primitive notion and as such is very powerful Example of spherical symmetry Consider a featureless beach ball that can be rotated about a vertical axis that passes through its center. The observer closes his eyes and we rotate the sphere. Then, when the observer opens his eyes, he cannot tell whether the sphere has been rotated or not. We conclude that the sphere has rotational symmetry about the rotation axis. featureless sphere rotation axis observer Rotational symmetry (23-10)

Line of Charge From SYMMETRY E is Radial and Outward

Infinite Sheet of Charge cylinder We got this same result from that ugly integration!

S1 S2 S3 (23-15)

S S' A A' (23-16)

(23-17)

Insulators In an insulator all of the charge is bound. None of the charge can move. We can therefore have charge anywhere in the volume and it can’t “flow” anywhere so it stays there. You can therefore have a charge density inside an insulator. You can also have an ELECTRIC FIELD in an insulator as well.

Recipe for applying Gauss’ Law 1. Make a sketch of the charge distribution 2. Identify the symmetry of the distribution and its effect on the electric field 3. Gauss’ law is true for any closed surface S. Choose one that makes the calculation of the flux  as easy as possible. 4. Use Gauss’ law to determine the electric field vector (23-13)

A uniformly charged cylinder. s s R