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

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.

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

“Love, which is the essence of God, is not for levity, but for the total worth of man.”
—Ralph Waldo Emerson (1803–1882)

“Don’t get involved in partial problems, but always take flight to where there is a free view over the whole single great problem, even if this view is still not a clear one.”
—Ludwig Wittgenstein (1889–1951)

“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)

“We assume that politicians are without honor. We read their statements trying to crack the code. The scandals of their politics: not so much that men in high places lie, only that they do so with such indifference, so endlessly, still expecting to be believed. We are accustomed to the contempt inherent in the political lie.”
—Adrienne Rich (b. 1929)