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:

“The only way into truth is through one’s own annihilation; through dwelling a long time in a state of extreme and total humiliation.”
—Simone Weil (1909–1943)

“Freedom of enterprise was from the beginning not altogether a blessing. As the liberty to work or to starve, it spelled toil, insecurity, and fear for the vast majority of the population. If the individual were no longer compelled to prove himself on the market, as a free economic subject, the disappearance of this freedom would be one of the greatest achievements of civilization.”
—Herbert Marcuse (1898–1979)

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

“There was books too.... One was “Pilgrim’s Progress,” about a man that left his family it didn’t say why. I read considerable in it now and then. The statements was interesting, but tough.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)