**Physics 230 Capacitor exercise with dielectrics**

Question: How do we get the biggest capacitance if we have enough dielectric to fill half of the capacitor volume? (It is a parallel plate capacitor.)

The two main choices look like this:

In the first case the capacitor may be modelled as two capacitors in series.
The individual capacitances
are:

Thus
the total capacitance is given
by:

and
thus

In
the second case our model is two capacitors in parallel. The individual
capacitances
are

So
the total capacitance
is

To
find which is the biggest we must compare
and
Is

Equivalently,
is

Yes!
Thus the capacitor on the right has the greatest capacitance.