A) ![U_0 = \frac{\epsilon_0 A V^2}{2d}](https://tex.z-dn.net/?f=U_0%20%3D%20%5Cfrac%7B%5Cepsilon_0%20A%20V%5E2%7D%7B2d%7D)
B) ![U_1=\frac{3\epsilon_0 A V^2}{2d}](https://tex.z-dn.net/?f=U_1%3D%5Cfrac%7B3%5Cepsilon_0%20A%20V%5E2%7D%7B2d%7D)
C) ![U_2=\frac{k\epsilon_0 A V^2}{2d}](https://tex.z-dn.net/?f=U_2%3D%5Cfrac%7Bk%5Cepsilon_0%20A%20V%5E2%7D%7B2d%7D)
Explanation:
A)
First of all, the capacitance of a parallel-plate capacitor filled with air is given by
(1)
where
is the vacuum permittivity
A is the area of the plates
d is the separation between the plates
The energy stored by a capacitor is given by
(2)
where
C is the capacitance
V is the potential difference across the plates
Substituting (1) into (2), we find an expression of the tenergy stored:
![U_0 = \frac{\epsilon_0 A V^2}{2d}](https://tex.z-dn.net/?f=U_0%20%3D%20%5Cfrac%7B%5Cepsilon_0%20A%20V%5E2%7D%7B2d%7D)
B)
In this part of the problem, the capacitor is disconnected from the battery.
This means that now the charge on the capacitor remains constant. The charge can be written as
![Q=CV](https://tex.z-dn.net/?f=Q%3DCV)
Since the charge is the same as in part A), we can write it explicitely:
(1)
We can write the energy stored in the capacitor using another equation:
(3)
In this case, the distance between the plates is increased to 3d, so the new capacitance is
(2)
Substituting (1) and (2) into (3), we find the new energy stored:
![U_1 = \frac{(\frac{\epsilon_0 A V}{d})^2}{2(\frac{\epsilon_0 A}{3d})}=\frac{3\epsilon_0 A V^2}{2d}](https://tex.z-dn.net/?f=U_1%20%3D%20%5Cfrac%7B%28%5Cfrac%7B%5Cepsilon_0%20A%20V%7D%7Bd%7D%29%5E2%7D%7B2%28%5Cfrac%7B%5Cepsilon_0%20A%7D%7B3d%7D%29%7D%3D%5Cfrac%7B3%5Cepsilon_0%20A%20V%5E2%7D%7B2d%7D)
3)
In this case, the capacitor is reconnected to the battery, so the potential difference is now equal to the initial potiential difference V.
In this case, however, a dielectric plate is moved inside the space between the plates. Therefore, the capacitance becomes
![C=\frac{k\epsilon_0 A}{d}](https://tex.z-dn.net/?f=C%3D%5Cfrac%7Bk%5Cepsilon_0%20A%7D%7Bd%7D)
where
k is the dielectric constant of the dielectric material
To calculate the energy stored, we can use again the original formula
![U=\frac{1}{2}CV^2](https://tex.z-dn.net/?f=U%3D%5Cfrac%7B1%7D%7B2%7DCV%5E2)
And substituting C and V, we find
![U_2=\frac{k\epsilon_0 A V^2}{2d}](https://tex.z-dn.net/?f=U_2%3D%5Cfrac%7Bk%5Cepsilon_0%20A%20V%5E2%7D%7B2d%7D)