Now recall that flux is the scalar product of a vector field and a bit of
surface
where
is some vector field and
is a surface with the direction defined by the normal to the surface. For a
series of connected surfaces
the total flux through the combined surface would be the sum of the individual
elements. For a vector field
passing through the surface this leads to
or when we go to infinitesimal areas
Now lets consider a charge
q
in the middle of a sphere
but
then
Φ
=
4
π
k
q
So for this case we get
We can generalize this to any closed surface. It is clear that for an
arbitrary closed source, we can draw a sphere around the source within the
arbitrary surface.. Think of bullets being fired from a gun, it is clear that
the bullets originating in the inner sphere all pass through the outer surface
and so one would expect that the flux would be the same. For example consider
to be a patch on the inner sphere and
to be its projection onto the outer arbitrary surface (with its normal making
an angle
θ
with
respect to the normal to
On the inner patch
and at the outer patch
So the two have equivalent fluxes.
Any electric field is the sum of fields of its individual sources so we can
write
or for charge distributed throughout the volume
Now we can apply Gauss' Theorem
The equation
must be true for any volume of any size, shape or location.
The only way that can be true is if:
Initially one may think that this is a much less clear way of posing Gauss'
Law. In practice it is much more useful than the integral form. Given an
arbitrary distribution of charge we can calculate the electric field anywhere
in space.
We can consider the same arguments for magnetic fields however there is one
major difference! There are no isolated source of magnetism. That is there are
no magnetic monopoles. This is an experimental fact. In fact people continue
to search for them but they have never been found. (Finding one would almost
certainly be a discovery worthy of a Nobel Prize). So we
have
or
The magnetic flux is
A
device that can maintain a potential difference, despite the flow of current
is a source of electromotive force. (EMF) The definition is mathematically
Faraday's
law states
thatThis
is simply an experimental fact.
Faraday's law could also be
written
I will leave it as a problem to show that this can also be
written
For a steady current flowing through a straight wire, the magnetic field at a
point at a perpendicular distance
r
from the wire, has a value
If
we integrate around the wire in a circle, then clearly we get
This
is true for irregular paths around the wire
but
for small
dl
d
l
cosθ
=
rdφ
In fact instead of current we use the surface integral of the current density
J
,
which is the current per unit area
Maxwell's great insight was to realize that this was incomplete. He
reasoned that
gives a
field so we should expect that
gives a
field.
Think
of a capacitor in a simple circuit. We can draw a surface such as shown in the
figure, with "surface 1" and take the line integral around the edge of the
surface. Now look at surface 2, this will have the same line integral, but
now the surface integral will be different. Clearly there is something
incomplete with Ampere's law as formulated above. Maxwell re wrote Ampere's
law
which
solves the problem.
Again it is left as an exercise to show that
Lets recall Maxwell's equations (in free space) in differential form
