Re: What exactly is a 'Weber' in terms of 'how much' magnetic flux it is?

This is a good question.

It is hard to explain the concept of magnetic
flux without using surface integral. However,
I'll give a try.

Suppose you put a mesh in flowing water. How
do you compute the water flowing through the 
mesh per second? The mesh is composed of many 
small hole. If you calculate the water flowing 
through each hole per second and sum them up,
you'll get the water flowing through the whole
mesh per second. Since each hole is so small,
it can be regarded as within a plane, therefore 
each hole has an area. Although the velocity of 
water may differ at different positions, since 
the hole is very small, the velocity of water 
flowing through a hole can be regarded as not changing
within the area of the hole. Therefore, if the velocity
of the water flowing through a hole is perpendicular
to the hole, the water flowing through the hole per second
is just V*A, where V is the velocity, A is the
area. However, if the angle of the velocity
and the normal vector of the hole (the normal vector
of the hole is a direction which is perpendicular
to the aperture of the hole and which is pointed
outward. You know the mesh has two sides. Before
you begin your calculation you should specify one
side as interior side and the other as exterior
side so that for each hole you can specify its
normal vector in a uniform way) is theta, the
water flowing through the hole per second is
V*A*cos(theta). This quantity (how much water flowing
through the hole per second) is called the velocity flux
through the hole. Some examples: if the water flowing
perpedicularlly out of the hole, the velocity flux is
V*A; if the water flowing perpendicularlly into the 
hole, the velocity flux is -V*A; if the water flowing 
parallelly to the hole, the velocity flux is 0, since
no water flowing in or flowing out. After the velocity
flux through each hole is calculated, the velocity flux
through the whole mesh is just a summation of the
velocity flux through all its holes.

Now we replace the velocity with magnetic intensity
(usually denoted as B, and is a vector field) in the
above descriptions, we'll get the definition of magnetic
flux. And in most cases the mesh is fictitious. Instead
of saying magnetic flux through a mesh, people say
magnetic flux through a surface. However, you know you can
divide the surface into meshes and calculate the magnetic
flux through each mesh cell and sum them up. Of couse
if you divide the surface in different ways you may
get different results, but if you divide it thinner and
thinner, you'll see such differences become smaller
and smaller. If you can divide the mesh into infinitely
small cells, difinitely you'll get one certain value,
that's the real value of magnetic flux. By dividing
the surface into infinitely small cells and summing
the magnetic flux through each cell up, actually you
have done a surface integral.

As to the "definition" of magnetic flux based on magnetic 
field lines, it's only an intutive description and can
not be used for accurate calculations.