Solutions to laplace's equation & uniqueness thrm #2 (Griffiths)

[iScience]

In griffith's intro to electrodynamics (4rth edition), ch. 3, pg. 121.

here is the second uniqueness thrm for the solutions to laplace's equation:

Second uniqueness thrm: In a volume V surrounded by conductors and containing a specified charge density ρ, the electric field is uniquely determined if the total charge on each conductor is given.

the only part I'm confused about is, in the beginning where he says "in a volume V surrounded by conductors and containing a specified charge density ρ"

Q: is the charge density ρ referring to that belonging to the region of space whose potential we're trying to determine? or is it referring to that of the conductor/insulator?

thanks

 

[WannabeNewton]

It is referring to that belonging to the volume V in which we are solving for the electric field.

 

[iScience]

thought so!

but then this leads to another confusion. I thought both uniqueness theorems were meant for the solutions to the laplace equation. The laplace equation however does not deal with charges inside the region.

so then, i guess, uniqueness thrm 1 is for both the laplace & the poisson equations while uniqueness thrm 2 is for just the poisson eqn?

please confirm this for me

 

[WannabeNewton]

Uniqueness theorem 2 is for Poisson's and uniqueness theorem 1 is for Laplace's.