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 think that if most guys in America could somehow get their fave-rave poster girl in bed and have total license to do whatever they wanted with this legendary body for one afternoon, at least 75 percent of the guys in the country would elect to beat her up.”
—Lester Bangs (1948–1982)

“In New York—whose subway trains in particular have been “tattooed” with a brio and an energy to put our own rude practitioners to shame—not an inch of free space is spared except that of advertisements.... Even the most chronically dispossessed appear prepared to endorse the legitimacy of the “haves.””
—Gilbert Adair, British author, critic. “Cleaning and Cleansing,” Myths and Memories (1986)

“I charge thee, Satan, housed within this man,
To yield possession to my holy prayers.”
—William Shakespeare (1564–1616)

“Nonwhite and working-class women, if they are ever to identify with the organized women’s movement, must see their own diverse experiences reflected in the practice and policy statements of these predominantly white middle-class groups.”
—Kimberly Crenshaw (b. 1959)