Partially Filled Capacitors I know that, for partially filled capacitors, one treats the space between the capacitor plates as two or more capacitors either in series or parallel.  However, I don't fully understand why the electric field set up inside the dielectric between the capacitor plates does not affect the field in the unfilled space.  Is it because the electric field contributions of the polarized atoms making up the dielectric slab completely cancel outside slab much the same way that the electric field contributions of the excess charges found on the two capacitor plates completely cancel outside the capacitor?
 A: When considering the effects of adding or removing dielectrics from a capacitor you must always decide whether you're leaving the capacitor connected to a battery, so holding the pd between the capacitor plates constant, or disconnecting (isolating) the plates, so that their charges are constant.
If the charges are held constant while a dielectric is inserted across the whole plate area, but only filling half the gap, it is indeed correct that the field in the air gap will stay the same. This can be discovered using charge, capacitance, pd relations for capacitors in series. More intuitively, the dielectric acquires unbalanced equal and opposite charges on its 'end' faces. The electric field due to a charged plane surface, at small distances away doesn't vary with distance away; the field is uniform. Therefore in the air gap, the fields due to the charges on either end-face of the dielectric are equal and opposite and cancel out, leaving just the field due to the charges on the plates! You may like to draw a diagram at this point.
So, answering your question:
"Is it because the electric field contributions of the polarized atoms making up the dielectric slab completely cancel outside slab much the same way that the electric field contributions of the excess charges found on the two capacitor plates completely cancel outside the capacitor?"
Yes! That's it!
If the pd between the plates is held constant, then inserting a dielectric across the whole plate area, but only filling half the gap, will increase the overall capacitance. Therefore more charge will flow on to the plates, and the field strength in the air gap will increase. Again, the fields in the air gap due to charges on the end faces of the dielectric cancel out. 
