Define dielectric constant in term of capacitance

Hi,

@zaarakhan48… and @kaavyamurali have given a suitable answer. Keep it up students.

Dielectric constant of the medium is the ratio of the electrostatic force of interaction between two given point charges held certain distance apart in vacuum/air to the force of interaction between the same two charges held the same distance apart in the material medium.

F₀/F_m=ε/ε₀=K……………………………………………(from Coloumb’s law)

We need to keep in mind that dielectrics are insulators and they do not conduct electricity however, charges can get induced in them.

Faraday observed that when a dielectric is fully occupying the space between the charged plates of a capacitor, its capacitance increases. The factor by which capacitance is multiplied depends on the nature of the dielectric introduced, and is called the dielectric constant (K) of the material.

K = capacitance of a capacitor with dielectric between the plates / capacitance of the same capacitor with vacuum/air in between the plates.

K = C_m/C₀

Where C_m is the capacitance when dielectric is between the plates and C₀ is the capacitance when air is between the plates.

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ELECTRIC FIELD ENERGY

Here's another way to think of the energy stored in a charged capacitor. If we consider the space between the plates to contain the energy (equal to 1/2 C V^2) we can calculate an energy DENSITY (Joules per volume). The volume between the plates is Area x distance between plates, or A d. Then the energy density (u) is

u = 1/2 C V^2 / A d = 1/2 eo E^2

where we have used C = Eo A / d and V =E d.

This is an important result because it tells us that
empty space can contain energy,
i.e., if there is an electric field in the "empty" space.

If we can get an electric field to travel (or propagate) through empty space
we can send or transmit energy!!!

Fig. 24.14 Effect of a dielectric between the plates of a parallel plate capacitor. The electrometer measures the potential difference. (a) With a given charge, the potential difference is Vo.
(b) With the same charge but with a dielectric between the plates, the potential difference V is smaller than Vo.

Most capacitors have an insulating (or DIELECTRIC) material between the plates. The presence of this dielectric INCREASES the capacitance of the capacitor compared to when the space between the plates was empty (a vacuum). Here's why: From measurements like the one above we know that insertion of a dielectric between the plates of a capacitor causes the potential difference Vo between the plates to decrease to V. The original capacitance, from Q = C V, is given by Co = Q / Vo and since Q must stay constant (it has nowhere to go!) we can write Q = Q, or Co Vo = C V, and if Vo decreases to V, Co must increase to C to keep the equation balanced. We can now define
K = C / Co = "dielectric constant" = the ratio of the capacitances. And from Co Vo = C V (above) we can write V = Vo Co / C = Vo / K. The dielectric constant (K) is a positive number equal to 1 for a vacuum and greater than 1 for other dielectric materials (Teflon - 2.1, glass = 7, water = 80, etc.), so we can say that the potential of a capacitor decreases by a factor of K when a dielectric material is added and the charge on the capacitor stays constant (i.e., the capacitor is not connected to a battery while the dielectric is inserted).

And since V = E d, we can conclude that the electric field between the plates must be reduced as a dielectric is inserted Q = (constant). See below for a physical picture of what is happening.

Fig. 24.15 (a) Electric field lines with vacuum between the plates.
(b) The induced charges on the faces of the dielectric decrease the electric field. (See figure below for a physical picture of how the (blue) induced charges on the faces of the dielectric are established.)

The electric field between the capacitor plates is reduced by the presence of the dielectric because the induced surface charges on the dielectric (see figure below) cause an electric field in the opposite direction of the original field in the charged capacitor. These fields tend to cancel each other resulting in a reduction of the original field.

Molecules in the dielectric material have their positive and negative charges separated slightly, causing the molecules to be oriented slightly in the electric field of the charged capacitor.

Fig. 24.20 Polarization of a dielectric in an electric field gives rise to thin layers of bound charges on the surfaces, creating positive and negative surface charge densities. The sizes of the molecules are greatly exaggerated for clarity.


Fig. 24.21 (a) Electric filed of magnitude Eo between two charged plates. (b) Introduction of a dielectric of dielectric constant K. (c) the induced surface charges and their field (thinner lines). (d) Resultant filed of magnitude Eo/K when a dielectric is between charged plates.