Flux through a Curved Surface

The expression for the flux in equation (2) was obtained for a flat surface.

What is the expression for when the surface is curved? See the figure below.

For a curved surface such as the one shown above, the area vector at each point, in general, differs in both direction and magnitude from the area vectors at other points. Also, the flow (of water, etc.) occurs in all different directions and rates. We can then divide the curved surface into n small segments, so small that each segment can be viewed as being flat. At the ith segment, where i is an integer from 1 to n, the area vector is and the water flow is specified by . Then, by applying equation (2), the small flux through the ith segment is

where is the angle between the vectors and .

The total flux through the curved surface can be obtained by summing all the 's from i=1 to i=n.

(Remember, i is an integer.)

In the limiting case,

we obtain
(3)

Notice that this is an area integral. In general, evaluating this integral will be rather complicated, but in this module we will use it only in cases of simple geometry.

Once you're ready, go on to our discussion of flux through a Gaussian surface.


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