Capacitance of a plate capacitor increases

[galipop]

How can I show that the capacitance of a plate capacitor increases if a dielectric material fills the space between the plates?

 

[MaximumTaco]

$$C = \varepsilon \frac{A}{d}$$

The permittivity is what you're interested in. A dielectric material has a higher permittivity than $$\varepsilon_0$$, the permittivity of free space.

 

[Graeme Robinson]

The capacitance of a plate capacitor can be calculated using the formula C = εA/d, where ε is the permittivity of the material between the plates, A is the area of the plates, and d is the distance between the plates.
When a dielectric material is introduced between the plates, it increases the permittivity (ε) of the material between the plates. This is because dielectric materials have a higher permittivity than air.
As a result, the capacitance of the plate capacitor increases because the permittivity value in the formula has increased. This means that for the same area and distance between the plates, the capacitance will be higher when a dielectric material is present.
This can also be demonstrated by using the concept of electric field lines. When a dielectric material is inserted between the plates, it polarizes and creates an electric field in the opposite direction to the field created by the plates. This reduces the net electric field between the plates, thereby increasing the capacitance.
In summary, the presence of a dielectric material between the plates of a capacitor increases the capacitance by increasing the permittivity of the material and reducing the net electric field between the plates.