Gauss' Law states that the electric flux, ,
through a closed surface, S, is equal to the
amount of charged enclosed by (i.e. inside) the surface S.
- Why don't charges that are outside the surface, S, make a contribution to ?
- If the total charge inside a closed surface (S) is zero,
is the electric field everywhere on the surface zero?
Conductors (metals), Faraday cage
A unique feature of a charged piece of metal is that, no matter what its shape is and if current is zero, then the electric field inside the piece of metal has to be zero. Free charges in the metal go to the surfaces of the metal and arrange themselves so that the electric field is zero everywhere inside the metal. Electrostatics is a linear theory, so the equilibrium charge distribution is unique, ie. there is only one arrangement of the charges on the surface which produces zero electric field everywhere inside the metal.
If there is a component of the electric field at the
surface of a conductor which is parallel to the
surface, then charge would flow. At equilibrium there
is no current in electrostatics, so the electric
field at the surface of a conductor is perpendicular
to the surface of the metal, everywhere. ie. the
electric field is in the direction of the surface normal.
It is then possible to draw a small Gaussian surface
around a surface element and to show that,
Closed metal surface - Faraday cage
An interesting effect is that a closed metal surface,
no matter what it's shape, screens out external sources
of electric field. Many high tech experiments
are done inside a metal room, called a Faraday cage,
so that effects due to stray electric fields
are avoided. Stated another way, if there are no
charges inside a closed metal surface, then the
electric field inside the metal surface is
zero everywhere. This is a much more general
result than the shell theorems which apply only
to uniform spherical shells of charge. Proving this
result is beyond the level of this course, so
I have to use the dreaded phrase ``it can be
Demo: Faraday's ice pail - a test of Gauss' law