By now you are familiar with GAUSS' LAW in its integral form,

where is the electric charge enclosed within the closed surface . Except for the "fudge factor" , which is just there to make the units come out right, GAUSS' LAW is just a simple statement that electric field "lines" are continuous except when they start or stop

There is also a GAUSS' LAW for the *magnetic* field
;
we can write it the same way,

where in this case refers to the enclosed

Suppose now we apply GAUSS' LAW to a small rectangular
region of space where the *z* axis is chosen to be in the
direction of the electric field, as shown in
Fig. 22.1.^{22.1}
The flux of electric field *into* this volume
at the bottom is
.
The flux *out*
at the top is
;
so the *net*
flux *out* is just
.
The definition of the *derivative of **E** with respect to **z*
gives us
where the partial derivative is used in acknowledgement of
the possibility that *E*_{z} may also vary with *x* and/or *y*.
GAUSS' LAW then reads
.
What is
? Well, in such a small region there is
some approximately constant *charge density*
(charge per unit volume) and the volume of this region is
,
so GAUSS' LAW reads
or just
.
If we now allow for the possibility of electric flux entering
and exiting through the other faces (*i.e.*
may also have *x* and/or *y* components), perfectly analogous
arguments hold for those components, with the resultant
"outflow-ness" given by

where the GRADIENT operator is shown in its cartesian representation (in rectangular coordinates

We are now ready to write GAUSS' LAW in its compact
*differential* form,

and for the magnetic field, assuming

These are the first two of MAXWELL'S EQUATIONS.

Jess H. Brewer - Last modified: Wed Nov 18 12:30:50 PST 2015