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.