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 techniques of opening conversation are universal. I knew long ago and rediscovered that the best way to attract attention, help, and conversation is to be lost. A man who seeing his mother starving to death on a path kicks her in the stomach to clear the way, will cheerfully devote several hours of his time giving wrong directions to a total stranger who claims to be lost.”
—John Steinbeck (1902–1968)

“Thou madest loose grace unkind;
Gavest bridle to their words, art to their pace.
O Honour, it is thou
That makest that stealth, which Love doth free allow.”
—Torquato Tasso (1544–1595)

“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)

“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)