The potential due to a polorized distribution is given by:
[tex]V( \vec{r}) = \frac{1}{4 \pi \epsilon _{0}} \int _{V} \frac{ \hat{r} \cdot \vec{P} ( \vec{r}')}{r^{2}} dV [/tex]
After working some voodoo math, this is worked into the form [tex]V = \frac{1}{4 \pi \epsilon _{0}} \oint _{S} \frac{1}{r} \vec{P} \cdot d \vec{a} - \frac{1}{4 \pi \epsilon _{0}} \int _{V} \frac{1}{r}( \nabla \cdot \vec{P}) dV [/tex]
the dipole moment per unit length is [itex]\vec{p}= \vec{P} dV [/itex]
What exactly does P mean? I'm confused about the meaning of P, which seems like a good place to start asking questions. I feel the rest of my misunderstandings stem from not really "getting" what P actually IS.
It also describes the potential of the surface charge as [itex]\sigma = \vec{P} \cdot \hat{n}[/itex] and [itex] \rho _b = - \nabla \cdot \vec{P}[/itex] which I have little clue where it comes from.
As an example of how befuddled I am, take the problem "Find the electric field produced by a uniformly polarized sphere of radius R". The answer is [itex]\frac{P}{3 \epsilon _{0}} r cos \theta [/itex] but when I set the integral up like this: [tex]\frac{1}{4 \pi \epsilon _{0}} \int ^{2 \pi} _{0} \int ^{\pi} _{0} \frac{P R^{2} cos \theta sin \theta}{ \sqrt{R^{2}+z^{2}-2Rz cos\theta}} d \theta d \phi [/tex]
This integral is probably the most hideous thing I've ever seen (used wolfram).
I figure a "bound charge" is each term of the final integral [itex]V = \frac{1}{4 \pi \epsilon _{0}} \oint _{S} \frac{1}{r} \vec{P} \cdot d \vec{a} - \frac{1}{4 \pi \epsilon _{0}} \int _{V} \frac{1}{r}( \nabla \cdot \vec{P}) dV [/itex], the first being the surface charge, and the second being the volume charge, but this doesn't really help me understand what it actually means.
NB sorry for any mistakes in the math text, it's a laborious process to write this out on a computer, and I'm prone to making mistakes as it is.
Is there a program that will simulate electric/magnetic fields? Visual as well as numerical, it would be interesting to see the electric field generated by different objects and such, which would be impossible to solve analytically.
[tex]V( \vec{r}) = \frac{1}{4 \pi \epsilon _{0}} \int _{V} \frac{ \hat{r} \cdot \vec{P} ( \vec{r}')}{r^{2}} dV [/tex]
After working some voodoo math, this is worked into the form [tex]V = \frac{1}{4 \pi \epsilon _{0}} \oint _{S} \frac{1}{r} \vec{P} \cdot d \vec{a} - \frac{1}{4 \pi \epsilon _{0}} \int _{V} \frac{1}{r}( \nabla \cdot \vec{P}) dV [/tex]
the dipole moment per unit length is [itex]\vec{p}= \vec{P} dV [/itex]
What exactly does P mean? I'm confused about the meaning of P, which seems like a good place to start asking questions. I feel the rest of my misunderstandings stem from not really "getting" what P actually IS.
It also describes the potential of the surface charge as [itex]\sigma = \vec{P} \cdot \hat{n}[/itex] and [itex] \rho _b = - \nabla \cdot \vec{P}[/itex] which I have little clue where it comes from.
As an example of how befuddled I am, take the problem "Find the electric field produced by a uniformly polarized sphere of radius R". The answer is [itex]\frac{P}{3 \epsilon _{0}} r cos \theta [/itex] but when I set the integral up like this: [tex]\frac{1}{4 \pi \epsilon _{0}} \int ^{2 \pi} _{0} \int ^{\pi} _{0} \frac{P R^{2} cos \theta sin \theta}{ \sqrt{R^{2}+z^{2}-2Rz cos\theta}} d \theta d \phi [/tex]
This integral is probably the most hideous thing I've ever seen (used wolfram).
I figure a "bound charge" is each term of the final integral [itex]V = \frac{1}{4 \pi \epsilon _{0}} \oint _{S} \frac{1}{r} \vec{P} \cdot d \vec{a} - \frac{1}{4 \pi \epsilon _{0}} \int _{V} \frac{1}{r}( \nabla \cdot \vec{P}) dV [/itex], the first being the surface charge, and the second being the volume charge, but this doesn't really help me understand what it actually means.
NB sorry for any mistakes in the math text, it's a laborious process to write this out on a computer, and I'm prone to making mistakes as it is.
Is there a program that will simulate electric/magnetic fields? Visual as well as numerical, it would be interesting to see the electric field generated by different objects and such, which would be impossible to solve analytically.