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Capacitor with Dielectric

What is a dielectric

A dielectric material is an insulator in which there are no free electrons.

This means that a dielectric will not be able to conduct current.

Dielectric constant

Dielectric constant K can be written as a ratio of EpsilonPermittivity of the medium to Epsilon subscript oPermittivity of free space.

Dielectric

It can also be defined as the ratio of C the capacitance with dielectric as the medium and Co is the capacitance with air or vacuum between the plates.

Dielectric

Further it can also be defined as the ratio of Eo the electric field between the plates with air or vacuum as the medium and E is the electric field between the plates with dielectric as the medium.

Dielectric

Polar and non-polar materials

Materials can be categeorized into polar or non-polar materials.

Polar materials

Materials and called polar materials when the molecules have a net dipole moment. This means that the center of mass of the positive charge and the center of mass of the negative charge in the molecule are slightly seperated

Non-polar materials

Materials and called non-polar materials when the molecules do not have a net diapole moment. This means that the center of mass of the positive charge and the center of mass of the negative charge in the molecule concide and are not able to create a dipole moment

Atoms due to their spehrical symmetry are always non-polar

Capacitor with dielectric

If we place a dielectric between the parellel plates of a capacitor and give a charge to the capacitor then in the absence of a dielectric there is an electric field Eo. Say the distance between the plates is d and the thickness of the dielectric is t. Now, since positive charge is on one plate and negative charge is on the other plate, it is going to induce diapole moment in the molecules of the dielectric. If the molecules were initially non-polar they will have this induced however, if they already had a diapole moment they will realign according to the positive and negative charge on the plates, thus orienting themselves in the direction of the electric field. Now the left side of the dielectric will have a net negative charge and ther right side a net positive charge. And, as a result the direction of the induced electric field E' inside the dielctric will be from right to left.

So since there is an electric field to the left due to induction, the net electric field E inside the dielectric will reduce and it will be:

E = Eo - E'

However the direction of E will be the same as the direction of Eo

Effect of a dielectric on the capacitance of a capacitor

Let A be the area of each plate

In the absence of a dielectric the capacitance Co can be written as:

Once the dielectric having a thickness t is placed between the plates the capacitance will change.

Vthe potential between the plates will be:

Here in is the product of electric field Eo and the distance of space free of the dielectric, which is given by (d-t). Et is the product of the electric field Einside the dielectic and the thickness t of the dielectric.

Now Dielectric

therefore

we know that

therefore

or

From Gauss's Law we know that where is a the surface charge density, that is the charge per unit area

so we can say that if is the charge on each plate, that is negative on one and positive on the other, then

or

we know that

therefore

hence

we know that capacitance

hence

This is the formula of capacitance with a dielectric which is not completely filling the plates

From here we can see that if the dielectric completely fills the plates then t will equal d. When we subsitute this then we get

we have seen that

therefore

Now as K is always greater than 1 that is it emplies that

Or if we put a dielectric between the plates of a capacitor then its capacitance is going to increase

Dielectric strength

It is the maximum value of electric field that a dielectric will be able to tolerate without electric breakdown.

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