Lorentz Cavity in Uniformly Polarised Dielectric: Exam Question

[manofphysics]

I just have a small question regarding lorentz cavity:

Refer to a small lorentz cavity in a uniformly polarised dielectric. as shown in fig.
E_{ex}: External electric field.
E_{P}: Electric field in the uniformly polarised dielectric (when sphere has NOT been cut out)
E_{L}:Electric field due to surface charge on cavity
E_{near}:Field due to dipoles inside cavity.

Now, E=E_{ex}+E_{P}+E_{L}+E_{near}

But, does not E_{P} change if we cut out a sphere from the polarised dielectric?
Is it that we are neglecting the small change in E_{P} due to cut out sphere and our result is an useful approximation, but not exact?

Please, any help will be appreciated. I need to understand this for my term exam.

 

[alxm]

I'm not sure, but offhand: If you form the cavity adiabatically, then E_L would compensate for the change in E_P, wouldn't it? So I suspect that's the approximation involved.