Begin2.DVI

in a vector field and try to determine how the vector field punctures this surface.

Let the surface be divided into nsmall areas ∆Si and let F
i =F
(xi, y i, zi) denote

the value of the vector field associated with each surface element. The dot product

F
i·∆S
i=F
i·ˆen∆Sirepresents the projection of the vector F
ionto the normal to the

element ∆Simultiplied by the area of the element. Such a product is a measure of

the number of field lines which pass through the area ∆Siand is called a flux across

the surface boundary. The total flux across the surface is denoted by the flux integral

φ= limn→∞

∆Si→ 0

∑n

i=1

F
i·∆S
i=

∫∫

R

F
·dS
=

∫∫

R

F
·ˆendS (8 .5)

The surface area over which the integration is performed can be part of a surface

or it can be over all points of a closed surface. The evaluation of a flux integral

over a closed surface measures the total contribution of the normal component of

the vector field over the surface.

The term flux can mean flow in some instances. For example, let an imaginary

plane surface of area 1 square centimeter be placed perpendicular to a uniform

velocity flow of magnitude V 0 , such that the velocity is the same at all points over

the surface. In 1 second there results a column of fluid V 0 units long which passes

through the unit of surface area. The dimension of the flux integral is volume per

unit of time and can be interpreted as the rate of flow or flux of the velocity across

the surface.

In the above example, the flux was a quantity which is recognized as volume rate

of flow. In many other problems the flux is only a definition and does not readily have

any physical meaning. For example, the electric flux over the surface of a sphere

due to a point charge at its center is given by the flux integral

∫∫

R

E
·dS ,
where E
is

the electrostatic intensity. The flux cannot be interpreted as flow because nothing

is flowing. In this case the flux is considered as a measure of the density of the field

lines that pass through the surface of the sphere.

The value for the flux depends upon the size of the surface that is placed in

the vector field under consideration and therefore cannot be used to describe a

characteristic of the vector field. However, if an arbitrary closed surface is placed

in a vector field and the flux integral over this surface is evaluated and the result is

divided by the volume enclosed by the surface, one obtains the ratio of VolumeFlux .By

letting the volume and surface area of the arbitrary closed surface approach zero, the