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:

“I only know that a rook
Ordering its black feathers can so shine
As to seize me senses, haul
My eyelids up, and grant

A brief respite from fear
Of total neutrality.”
—Sylvia Plath (1932–1963)

“If I had not come to America, where I felt free to formulate tentatively insights at which I had empathically arrived, I would have accomplished very little. I would never have begun to publish, to teach, to undertake research. Because if one does not find an assenting echo to one’s ideas, if one is passed over, as I was in Vienna, then one cannot create. To create, after all, is to believe that what one says will count.”
—Margaret S. Mahler (1897–1985)

“I have never doubted your courage and devotion to the cause. But you have just lost a Division, and prima facie the fault is upon you; and while that remains unchanged, for me to put you in command again, is to justly subject me to the charge of having put you there on purpose to have you lose another.”
—Abraham Lincoln (1809–1865)

“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)