Divergence
The divergence of a vector at a point is defined to be
the total flux leaving a small volume surrounding the point, divided
by the volume.
Figure 1 The volume element
is shown.
|
| (1) |
Since,
by Gauss' law,
| (2) |
and since,
| (3) |
| (4) |
This is the differential form of Gauss' law.
Consider the small volume shown in Figure 1.
The total flux due to the x component of the vector E is,
| (5) |
This flux divided by the volume is,
| (6) |
| (7) |
Notice that for small
D
x ,
equation 7
is the definition of the partial derivative of Ex with respect to
x .
If you add the flux produced by the y and z components of E.
| (8) |
The del operator,
| (9) |
can be used to express the divergence of E.
| (10) |
Gauss' law in differential form is usually written as,
| (11) |
The Divergence Theorem
If the divergence from Equation 1 is multiplied by the volume element
,
| (12) |
The right hand side of Equation 12 is the flux out of the volume element
.
When Equation 12 is integrated over a volume, the total flux is the sum
of the flux out of each volume element
.
Flux between adjacent elements of
volume cancel.
For two adjacent volume elements,
the flux through their common surface is positive for one volume element and negative
for the other.
Only the flux through the surface of the total volume contributes to the integral.
| (13) |
where
S
is the surface of the volume
V.
The integral of the divergence of a vector over a volume is equal to the
integral of the vector over the surface of the volume. This is the divergence theorem.
More Electromagnetics
Other formulas from electromagnetic theory can be efficiently written using
the del operator. Since,
| (14) |
Gauss' law (Equation 11) becomes,
| (15) |
The equation in the box is Poisson's equation
where,
| (16) |
is the Laplacian operator and
r
is the charge density.
In a region where there is no charge, r
is zero
and Poisson's equation becomes Laplace's equation.
| (17) |
Laplace's equation is a second order partial differential equation that can be
solved to find the potential field when the values of the potential are
known at the boundaries.