#### dRic2

Gold Member

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- 102

- Homework Statement
- My professor told us to prove $$\int_{V} \mathbf x \rho_{b} d_3 \mathbf x = \int_{V} \mathbf P d_3 \mathbf x$$ using just the relation between bound charges and the electric dipole moment per unit volume.

- Homework Equations
- $$\rho_b = - \nabla \cdot \mathbf P$$

I spent a good amount of time thinking about it and in the end I gave up and asked to a friend of mine. He said it's a 1-line-proof: just "integrate by parts" and that's it. I'm not sure you can do that, so instead I tried using the identity:

to express the first term on the right-hand side (which is my ##\mathbf x \nabla \cdot \mathbf P##) as a summation of all the other therms, but then I don't know how to get rid of all those integrals...

PS: I have to do the same thing for the magnetic field (prove that ##\int_{V} \mathbf x × \mathbf J = \int_{V} \mathbf M ##) but I hope that, once I get the trick, I can adapt it to the ather case.

Thanks

Ric

to express the first term on the right-hand side (which is my ##\mathbf x \nabla \cdot \mathbf P##) as a summation of all the other therms, but then I don't know how to get rid of all those integrals...

PS: I have to do the same thing for the magnetic field (prove that ##\int_{V} \mathbf x × \mathbf J = \int_{V} \mathbf M ##) but I hope that, once I get the trick, I can adapt it to the ather case.

Thanks

Ric