# Gauss Law

**Topics:**Electric field, Electromagnetism, Electrostatics

**Pages:**8 (1084 words)

**Published:**November 24, 2011

AP Physics C

Electric Flux

Let's start be defining an area on the surface of an object. The magnitude is “A” and the direction is directed perpendicular to the area like a force normal.

A

E

Flux ( or FLOW) is a general term associated with a FIELD that is bound by a certain AREA. So ELECTRIC FLUX is any AREA that has a ELECTRIC FIELD passing through it.

We generally define an AREA vector as one that is perpendicular to the surface of the material. Therefore, you can see in the figure that the AREA vector and the Electric Field vector are PARALLEL. This then produces a DOT PRODUCT between the 2 variables that then define flux.

Electric Flux

The electric field lines look like lines of a "fluid". So you can imagine these lines are flowing (even though nothing is really flowing). The word FLUX roughly means FLOW. So based on this idea we can define the ELECTRIC FLUX as the ELECTRIC FEILD through a SURFACE AREA. Since the area vector is defined as perpendicular to the surface and the electric field goes through it, we define this equation as a dot product, similar to the work function.

Φ E = E • A = EA cos θ dΦ = ∫ Eda

A differential amount of flux is the cross product between the electric field and a differential amount of area. Since you want the total flux, you integrate to sum up all the small areas. Thus the TOTAL FLUX is found by integrating over the ENTIRE SURFACE. The circle on the integration sign simply means the surface is CLOSED!!.

∫

Electric Flux

Visually we can try to understand that the flux is simply the # of electric field lines passing through any given area. In the left figure, the flux is zero.

In the right figure, the flux is 2.

• When E lines pass outward through a closed surface, the FLUX is positive • When E lines go into a closed surface, the FLUX is negative

Electric Flux

What is the electric flux of this cylinder?

Φ E = ∑ Φ = Φ1 + Φ 2 + Φ 3 , E = constant, A1 = A2 Φ E = EA1 cos 0 + EA2 cos180 + EA3 cos 90 Φ E = EA1 (1) + EA2 (−1) + 0 ΦE = 0 What does this tell us?

This tells us that there are NO sources or sinks INSIDE the cylindrical object.

Gauss’ Law

Where does a fluid come from? A spring! The spring is the SOURCE of the flow. Suppose you enclose the spring with a closed surface such as a sphere. If your water accumulates within the sphere, you can see that the total flow out of the sphere is equal to the rate at which the source is producing water.

In the case of electric fields the source of the field is the CHARGE! So we can now say that the SUM OF THE SOURCES WITHIN A CLOSED SURFACE IS EQUAL TO THE TOTAL FLUX THROUGH THE SURFACE. This has become known as Gauss' Law

Gauss’ Law

The electric flux (flow) is in direct proportion to the charge that is enclosed within some type of surface, which we call Gaussian.

The vacuum permittivity constant is the constant of proportionality in this case as the flow can be interrupted should some type of material come between the flux and the surface area. Gauss’ Law then is derived mathematically using 2 known expressions for flux.

∫ E • dA =

qenc

εo

Gauss’ Law – How does it work?

Consider a POSITIVE POINT CHARGE, Q.

Step 1 – Is there a source of symmetry?

Yes, it is spherical symmetry! You then draw a shape in such a way as to obey the symmetry and ENCLOSE the charge. In this case, we enclose the charge within a sphere. This surface is called a GAUSSIAN SURFACE.

Step 2 – What do you know about the electric field at all points on this surface? It is constant.

E ∫ da =

qenc

The “E” is then brought out of the integral.

εo

Gauss’ Law – How does it work?

Step 3 – Identify the area of the Gaussian surface? In this case, summing each and every dA gives us the surface area of a sphere.

E (4πr ) =

2

qenc

εo

Step 4 – Identify the charge enclosed? The charge enclosed is Q!

E (4πr ) =

2

Q

εo

→

Q E= 4πr 2ε o

This is the...

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