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:

    The total and universal want of manners, both in males and females, is ... remarkable ... that polish which removes the coarser and rougher parts of our nature is unknown and undreamed of.
    Frances Trollope (1780–1863)

    To be free in an age like ours, one must be in a position of authority. That in itself would be enough to make me ambitious.
    Ernest Renan (1823–1892)

    I never thought that the possession of money would make me feel rich: it often does seem to have an opposite effect. But then, I have never had the opportunity of knowing, by experience, how it does make one feel. It is something to have been spared the responsibility of taking charge of the Lord’s silver and gold.
    Lucy Larcom (1824–1893)

    If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.
    —J.L. (John Langshaw)