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:

“For, the expectation of gratitude is mean, and is continually punished by the total insensibility of the obliged person. It is a great happiness to get off without injury and heart-burning, from one who has had the ill luck to be served by you. It is a very onerous business, this being served, and the debtor naturally wishes to give you a slap.”
—Ralph Waldo Emerson (1803–1882)

“Our Germany’s dead. However hard this may be for some of us older people, it’s a blessing for our children. Our children grew up against new backgrounds, new horizons. And they are free. Free to grow up as children. Free to run and to laugh without being forced into uniforms. Without being forced to march up and down streets, singing battle songs.”
—Emeric Pressburger (1902–1988)

“Those things for which the most money is demanded are never the things which the student most wants. Tuition, for instance, is an important item in the term bill, while for the far more valuable education which he gets by associating with the most cultivated of his contemporaries no charge is made.”
—Henry David Thoreau (1817–1862)

“Dogmatic theological statements are neither logical propositions nor poetic utterances. They are “shaggy dog” stories; they have a point, but he who tries too hard to get it will miss it.”
—W.H. (Wystan Hugh)