I would like to share an elementary introduction to Gauss law
shown below. Feel free to remove my name and make any
changes you wish. Comments, if any, will be appreciated.
The terms "lines flowing into the box" and "lines flowing
out of the box" were invented to promote discussion, and
to emphasize that nothing is really flowing unless sources
or sinks of E become sources or sinks of v fields.
Ludwik Kowalski
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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:
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.