Electromagnetic Fields: Divergence

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ELECTROMAGNETIC FIELDS AND THEORY: FIELDS AND VECTORS: Divergence

1.6.2 Divergence.

This is another operator. Its argument is a vector function y it’s result is a scalar function. Using the following expression, the divergence of vector function “E” is indicated.

(40)

the same one can be defined using the concept of flow in the following way,

(41)

this expression can be translated into words the following way: in a particular point, the divergent function is equal to the limit of the flow that passes through the enclosed surface SO from inside to outside, divided by the volume, when the enclosed volume by surface SO and that contains the point tends to zero.

As an example we will consider the vector field, constituted by the velocity of water that moves throughout a canal and a imaginary enclosed surface SO that’s entirely under water. In normal conditions, as much water enters in the enclosed region SO as comes out; this will mean that there is no net flow from enclosed region SO. It also means that (if conditions are maintained in the limit), that the divergence of velocity is zero.

If in this example there where a source of water within the enclosed region SO, the out coming net flow of the velocity would be positive and consequently, the divergence in this point would be positive. If in another part of the enclosed region by SO, a drain could be found, the out coming net flow would be negative, as well as the divergence at that given point.

In the mentioned systems of coordinates, the divergence is given by the following expressions,

Rectangular:

        (42)

Cylindrical:

        (43)

Spherical:

        (44)