Answer:
a) The ratio is .
b) The total stored energy increases after the dielectric is inserted.
Explanation:
<em><u>Before the dielectric is inserted:</u></em>
We have two parallel plate capacitors ( and ), each with a capacitance C, connected in series to a battery that has voltage V.
The two capacitors can be replaced by an equivalent one. The equivalent capacitance for two capacitors in series is
The potential differences across the two capacitors are:
and
The total potential difference is the sum of the potential differences across each capacitor:
The initial stored energy is:
Using instead we have that .
<u><em>After the dielectric is inserted:</em></u>
A dielectric with is inserted between the plates of one of the capacitors while the two capacitors are connected to the battery. The total potential difference reamins the same. If Q is the charge in the plates without the dielectric, in the capacitor with the dielectric the charge increases to and the capacitance to .
Now we have that the equivalent capacitance is:
So the stored energy after the dielectric is inserted becomes:
- So, the ratio is:
- Because if this means that the total stored energy increases after the dielectric is inserted.