Bound Charge Density

I thought about this relation for a really long time.

I came to my own conclusion, but if it’s a ridiculous interpretation, I’ll just decline the arrows. I’d be grateful if you shoot me poison arrows.

It seems to have been tough because the concept of P is something I’m encountering for the first time.

First, P represents the electric dipole moment per unit volume, and now the moment I look at this P, ““aha~ in a ‘unit volume’ there are charges of opposite sign at opposite positions!!!””

I decided to think of it this way.

Then can I treat the magnitude of P, |P|, as the amount of charge piled up on one side of the unit volume?!?!

For instance, if the magnitude of P is q, then “aha~ on one side of the unit volume there’s q, and on the other side there’s -q”

“Ah, but since P is per unit volume, rather than q, it would be more appropriate to write it as sigma”

Let’s say a lil’ volume body like this is uniformly polarized.

Then can I think of it as thiss shape~

In that case, when one side’s area is S, let’s say P tiiimes S is Q.

(In the end, dividing Q by the area gives charge per unit area, and that is precisely the magnitude of P…?!)

Now then, if we say it ain’t sliced so prettyyy~ like that,

It’d be this lil’ shape! Here, in order to get sigma, the charge per unit area, we have to divide the total charge by the slanted area

There’s a bit of a leap or a stretch here.

The point that you can’t jus’ blindly think of P as sigma is seepin’ in, but

now

let’s investigate this equation.

Let’s think that P means two charges of different signs ‘are’ at opposite positions

rho_b is the charge density inside (in) the material. Summing up all the charge densities for the unit volumes on the inside gives

now now now now now now let’s see

If there’s (-) on the inside then there will be (+) on the outside, and (+) being on the inside means (-) is on the outside.

The sum of all the charges on the inside equals the sum of all the charges on the outside with a minus sign slapped on, so the two are equal!?!?

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That is

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Originally written in Korean on my Naver blog (2014-11). Translated to English for gdpark.blog.