Chronology | Current Month | Current Thread | Current Date |
[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
SENDING IT AGAIN. IT SHOULD LOOK(vector), and
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
flux F (scalar). The first two, and the concept of field lines, arefamiliar.
Before learning about the formal definition of flux consider thefollowing.
Suppose that a single charged particle is hidden in a box. It can not beseen
but the electric field lines coming from the box are "flowing out of thebox"
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 lineswould
be "flowing into the box." Note that we are able to say something aboutQ on
the basis of the field lines.be
Suppose an identical box with ten times larger Q was presented. Would we
able to distinguish it from the previous box? Certainly; the linedensity (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, bysaying
convention, is proportional to E in any given region. Note we are again
something about charges hidden from us on the basis of fields theyproduce
outside of an enclosed region. We also know that the line densitieswould
decrease if boxes were larger while the total number of lines, throughthe
entire closed surface, would remain the same. The concept of flux helpsus to
deal with hidden charges. And, as we will see, to calculate E in someour
situations in which information about enclosed charges is given.
Suppose a very small areas, dS, is considered anywhere on the surface of
imaginable box. Let us represent dS by a vector pointing away from thebox and
always perpendicular to the surface. The electric field, E, in themiddle of
our tiny surface element may or may not be parallel to the dS. But italways
has a parallel component equal to E*cos(THETA), where THETA is the angleE*dS*cos(THETA).
between E and dS. The flux element dF is defined as a product
Those familiar with vector algebra would probably prefer to say it is adot
product of two vectors, E and dS. But that is not essential. Note thatwhen the
cos(THETA) is negative when THETA is larger that 90 degrees, that is
direction of E is "into the box".not
Also note that our imaginable closed surface can have any shape; it does
have to be a rectangular box. In principle nothing prevents us fromelements,
subdividing the entire surface area into a large number of tiny dS
to calculate the dF at each of them and add all flux elements dF. Bydoing
this we would calculate the total flux F. By definition the sum of dF isthe
total flux F. Here is the most simple illustration, it is a sphericalbox with
a charge Q in the center. It is clear that in this case all vectors Eare
identical in magnitude and that each cos(THETA) is either +1 or ?1. Forthat
reason the total flux is simply the product E and the area of our sphereour
(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
spherical box" (negative Q).Q at
We want to calculate F for this simple case, that is for a point charge
the center of a sphere of radius r. The magnitude of E at any surfaceelement
is E=k*Q/r^2, where k=1/(4*Pi*eps_o), as usual. This leads to:sphere
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
of any radius. Also notice that F is positive when Q is positive andnegative
when Q is negative. This remarkable relation between a single charge Q,any
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
kind, for example, egg-shaped, and for an arbitrary location of a chargein
inside that surface. The formal mathematical proof of this can be found
many textbooks. Next consider several charges, Q1, Q2, Q3, etc. locatedinside
an enclosed region. In that casemany
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
charges, provided Q is the algebraic sum of individual charges. Chargesinstead of
outside an enclosed region do not contribute to the net flux through a
surrounding closed surface. Some people prefer to say Gauss "law"
Gauss "theorem"; both references are acceptable.