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:

“Computers are good at swift, accurate computation and at storing great masses of information. The brain, on the other hand, is not as efficient a number cruncher and its memory is often highly fallible; a basic inexactness is built into its design. The brain’s strong point is its flexibility. It is unsurpassed at making shrewd guesses and at grasping the total meaning of information presented to it.”
—Jeremy Campbell (b. 1931)

“You have waited, you always wait, you dumb, beautiful ministers,
We receive you with free sense at last, and are insatiate
hence-forward,
Not you any more shall be able to foil us, or withhold yourselves
from us,
We use you, and do not cast you aside—we plant you permanently within us,
We fathom you not—we love you—there is perfection in you also,
You furnish your parts, toward eternity,
Great or small, you furnish your parts toward the soul.”
—Walt Whitman (1819–1892)

“To fight aloud is very brave,
But gallanter I know,
Who charge within the bosom
The Cavalry of Woe.”
—Emily Dickinson (1830–1886)

“The true critic is a scrupulous avoider of formulae; he refrains from statements which pretend to be literally true; he finds fact nowhere and approximation always.”
—T.S. (Thomas Stearns)