Gauss's Law - Equivalence of Total and Free Charge Statements

Equivalence of Total and Free Charge Statements

Proof that the formulations of Gauss's law in terms of free charge are equivalent to the formulations involving total charge.
In this proof, we will show that the equation

is equivalent to the equation

Note that we're only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the divergence theorem.

We introduce the polarization density P, which has the following relation to E and D:

and the following relation to the bound charge:

Now, consider the three equations:

The key insight is that the sum of the first two equations is the third equation. This completes the proof: The first equation is true by definition, and therefore the second equation is true if and only if the third equation is true. So the second and third equations are equivalent, which is what we wanted to prove.

Read more about this topic:  Gauss's Law

Famous quotes containing the words total, free, charge and/or statements:

    Unlike Descartes, we own and use our beliefs of the moment, even in the midst of philosophizing, until by what is vaguely called scientific method we change them here and there for the better. Within our own total evolving doctrine, we can judge truth as earnestly and absolutely as can be, subject to correction, but that goes without saying.
    Willard Van Orman Quine (b. 1908)

    The poet will prevail to be popular in spite of his faults, and in spite of his beauties too. He will hit the nail on the head, and we shall not know the shape of his hammer. He makes us free of his hearth and heart, which is greater than to offer one the freedom of a city.
    Henry David Thoreau (1817–1862)

    What art thou that usurp’st this time of night,
    Together with that fair and warlike form
    In which the majesty of buried Denmark
    Did sometimes march? By heaven I charge thee speak!
    William Shakespeare (1564–1616)

    He admired the terrible recreative power of his memory. It was only with the weakening of this generator whose fecundity diminishes with age that he could hope for his torture to be appeased. But it appeared that the power to make him suffer of one of Odette’s statements seemed exhausted, then one of these statements on which Swann’s spirit had until then not dwelled, an almost new word relayed the others and struck him with new vigor.
    Marcel Proust (1871–1922)