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GAUSS' LAW

The EQUATION OF CONTINUITY (see above) describes the conservation of "actual physical stuff" entering or leaving an infinitesimal region of space dV. For example, $\Vec{J}$ may be the current density (charge flow per unit time per unit area normal to the direction of flow) in which case $\rho$ is the charge density (charge per unit volume); in that example the conserved "stuff" is electric charge itself. Many other examples exist, such as FLUID DYNAMICS (in which mass is the conserved stuff) or HEAT FLOW (in which energy is the conserved quantity). In ELECTROMAGNETISM, however, we deal not only with the conservation of charge but also with the continuity of abstract vector fields like $\Vec{E}$ and $\Vec{B}$. In order to visualize $\Vec{E}$, we have developed the notion of "electric field lines" that cannot be broken except where they originate (from positive charges) and terminate (on negative charges). [This description only holds for static electric fields; when things move or otherwise change with time, things get a lot more complicated . . . and interesting!] Thus a positive charge is a "source of electric field lines" and a negative charge is a "sink" - the charges themselves stay put, but the lines of $\Vec{E}$ diverge out of or into them. You can probably see where this is heading.

GAUSS' LAW states that the net flux of electric field "lines" out of a closed surface ${\cal S}$ is proportional to the net electric charge enclosed within that surface. The constant of proportionality depends on which system of units one is using; in SI units it is $1/\epsilon_\circ$. In mathematical shorthand, this reads

$$\epsilon_\circ \; \oSurfIntS \Vec{E} \cdot d\Vec{A} \; = \; Q_{\rm encl} .$$

Recalling our earlier discussion of DIVERGENCE, we can think of $\Vec{E}$ as being a sort of flux density of conserved "stuff" emitted by positive electric charges. Remember, in this case the charges themselves do not go anywhere; they simply emit (or absorb) the electric field "lines" which emerge from (or disappear into) the enclosed region. The rate of generation of this "stuff" is $Q_{\rm encl}/\epsilon_\circ$. We can then apply GAUSS' LAW to an infinitesimal volume element using Fig. 1 with $\epsilon_\circ \Vec{E}$ in place of $\Vec{J}$. Except for the "fudge factor" $\epsilon_\circ$ and the replacement of $\dot{Q}$ by $Q_{\rm encl}$, the same arguments used to derive the EQUATION OF CONTINUITY lead in this case to a formula relating the divergence of $\Vec{E}$ to the electric charge density $\rho$ at any point in space, namely

$$\fbox{\hbox{$\displaystyle \Div{E} \; = \; {1 \over \epsilon_\circ} \; \rho $}} .$$

This is the differential form of GAUSS' LAW.

Poisson and Laplace

Even in its differential form, GAUSS' LAW is a little tricky to solve analytically, since it is a vector differential equation. Generally we have an easier time solving scalar differential equations, even though they may involve higher order partial derivatives. Fortunately, we can convert the former into the latter: recall that the vector electric field can always be obtained from the scalar electrostatic potential using

$$\Vec{E} \; \equiv \; - \Grad{\phi} .$$

Thus $\hbox{\rm div} \Vec{E} \equiv \Div{E} = - \Grad{} \cdot \Grad{\phi}$ or

$$\fbox{\hbox{$\displaystyle \Delsq{\phi} \; = \; - \; {1 \over \epsilon_\circ} \; \rho $}} .$$

This relation is known as POISSON'S EQUATION. Its simplified cousin, LAPLACE'S EQUATION, applies in regions of space where there are no free charges:

$$\fbox{\hbox{$\displaystyle \Delsq{\phi} \; = \; 0 $}} .$$

Each of these equations finds much use in real electrostatics problems. Advanced students of electromagnetism learn many types of functions that satisfy LAPLACE'S EQUATION, with different symmetries; since a conductor is always an equipotential (every point in a given conductor must have the same $\phi$, otherwise there would be an electric field in the conductor that would cause charges to move until they cancelled out the differences in $\phi$), empty regions surrounded by conductors of certain shapes must have $\phi$ with a spatial dependence satisfying those BOUNDARY CONDITIONS as well as LAPLACE'S EQUATION. One can often write down a complicated-looking formula for $\phi$ almost by inspection, using this favourite method of Physicists and Mathematicians, namely . . . KNOWING THE ANSWER.