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Re: GAUSS LAW



SENDING IT AGAIN. IT SHOULD LOOK
BETTER ON YOUR SCREEN NOW.

Introduction to Gauss Law

Ludwik Kowalski, spring 2001

Gauss Law is based on three concepts: charge Q (scalar), field E (vector), and
flux F (scalar). The first two, and the concept of field lines, are familiar.
Before learning about the formal definition of flux consider the following.
Suppose that a single charged particle is hidden in a box. It can not be seen
but the electric field lines coming from the box are "flowing out of the box"
through all six surfaces. What can we say about the sign of the charge? We
know it must be positive. For a negative charge the electric field lines would
be "flowing into the box." Note that we are able to say something about Q on
the basis of the field lines.

Suppose an identical box with ten times larger Q was presented. Would we be
able to distinguish it from the previous box? Certainly; the line density (how
many lines per unit area, or how close the lines are from each other) would be
larger for the second box. This is true because the line density, by
convention, is proportional to E in any given region. Note we are again saying
something about charges hidden from us on the basis of fields they produce
outside of an enclosed region. We also know that the line densities would
decrease if boxes were larger while the total number of lines, through the
entire closed surface, would remain the same. The concept of flux helps us to
deal with hidden charges. And, as we will see, to calculate E in some
situations in which information about enclosed charges is given.

Suppose a very small areas, dS, is considered anywhere on the surface of our
imaginable box. Let us represent dS by a vector pointing away from the box and
always perpendicular to the surface. The electric field, E, in the middle of
our tiny surface element may or may not be parallel to the dS. But it always
has a parallel component equal to E*cos(THETA), where THETA is the angle
between E and dS. The flux element dF is defined as a product E*dS*cos(THETA).
Those familiar with vector algebra would probably prefer to say it is a dot
product of two vectors, E and dS. But that is not essential. Note that
cos(THETA) is negative when THETA is larger that 90 degrees, that is when the
direction of E is "into the box".

Also note that our imaginable closed surface can have any shape; it does not
have to be a rectangular box. In principle nothing prevents us from
subdividing the entire surface area into a large number of tiny dS elements,
to calculate the dF at each of them and add all flux elements dF. By doing
this we would calculate the total flux F. By definition the sum of dF is the
total flux F. Here is the most simple illustration, it is a spherical box with
a charge Q in the center. It is clear that in this case all vectors E are
identical in magnitude and that each cos(THETA) is either +1 or ?1. For that
reason the total flux is simply the product E and the area of our sphere
(4*Pi*r^2). The product is positive when E lines are "flowing out of the
sphere" (that is when Q is positive) and negative when they "flow into our
spherical box" (negative Q).

We want to calculate F for this simple case, that is for a point charge Q at
the center of a sphere of radius r. The magnitude of E at any surface element
is E=k*Q/r^2, where k=1/(4*Pi*eps_o), as usual. This leads to:

Q Q
F = 4*Pi*r^2 * ---------------- = -----
4*Pi*eps_o*r^2 eps_o

Note that 4*Pi*r^2 canceled; it means that the theorem is valid for a sphere
of any radius. Also notice that F is positive when Q is positive and negative
when Q is negative. This remarkable relation between a single charge Q,
located in the center of a sphere, and the flux of the electric field E
through a surface of that sphere, is an illustration of Gauss theorem.

It turns out that the theorem can be generalized to a closed surface of any
kind, for example, egg-shaped, and for an arbitrary location of a charge
inside that surface. The formal mathematical proof of this can be found in
many textbooks. Next consider several charges, Q1, Q2, Q3, etc. located inside
an enclosed region. In that case

F=F1+F2+F3+etc. = Q1/eps_o + Q2/eps-o + Q3/eps_o + etc.

This shows that the first formulation of Gauss law remains valid for many
charges, provided Q is the algebraic sum of individual charges. Charges
outside an enclosed region do not contribute to the net flux through a
surrounding closed surface. Some people prefer to say Gauss "law" instead of
Gauss "theorem"; both references are acceptable.