Macroscopic Property of Dielectrics, Physics tutorial

Simple Model of the Dielectric Material:

It is noted that:

a) Each and every material is made up of extremely large number of atoms or molecules.

b) An atom comprises of a positively charged nucleus and negatively charged particles having electrons revolving around it.

c) The net positive charge of the nucleus is balanced via the net negative charge of the electrons in the atom, in such a way that the atom, as a whole, is electrically neutral with respect to any point present outside the atom.

d) A molecule might be comprised by atom of the similar type, or of different types.

To comprehend the polarization we shall consider a crude model of the atom. The nucleus is at the centre and different electrons revolving around it can be thought of as a spherically symmetric cloud of electrons. For points exterior the atom this cloud of electrons can be regarded as concentrated at the centre of the atom as the point charge.

In most of the molecules and atoms, the centers of negative and positive charges coincide with one other, while, in several molecules the centers of the two charges are positioned at dissimilar points. These molecules are termed as polar molecules.

Moreover, we noticed that in dielectrics, all the electrons are firmly bound to their particular atoms and are not able to move about freely. In the absence of an electric field, the charges within the molecules or atoms occupy their equilibrium places.

Behavior of a Dielectric in an Electric Field:

We are familiar that in a dielectric material, the centers of positive and negative charges of its atoms are found to coincide at the centre of the sphere. The charge experiences a force in the presence of an electric field.

Thus, whenever a dielectric material is positioned in an electric field, the positive charge of each and every atom experiences a force all along the direction of the field and the negative charge in the direction opposite to it. This result in small displacement of charge centers of the atoms or molecules. This is as well true of molecules whose charge centers don't coincide in the absence of an electric field. The separation of the charge centers due to an applied field 'E' is represented as follows:

1810_separation of the charge centres.jpg

This phenomenon is termed as polarization. Therefore if an electrically neutral molecule is positioned in an electric field, it gets polarized, having positive charges moving towards one end and negative charges towards the other. Or else neutral atom therefore becomes a dipole having a dipole moment that is proportional to the electric field. 

Now we consider the other type of molecule in which the charge centers don't coincide:

Because of this reason the molecule already acquires a dipole moment. These materials are termed as polar materials. For these materials, let the initial orientation of the dipole axis be AOB as represented in the figure given below:

831_Molecule possessing a dipole moment.jpg

Now an electric field 'E' is applied. This field pulls the charge centers all along the lines parallel to its direction. Therefore the electric field applies a torque on the dipole causing it to reorient in the direction of the field. In the absence of an electric field such polar materials don't have any resultant dipole moment, as the dipoles of dissimilar molecules are oriented in random directions due to the thermal agitation. If an electric field is applied, each of such molecules reorients itself in the direction of the field, and a total polarization of the material outcomes. The reorientation or polarization of the medium is not perfect again due to the thermal agitation. Therefore polarization based both on field (linearly) and temperature.

Non-Polar and Polar Molecules

There are basically two kinds of molecules. One in which the centre of positive charges coincide by the centre of the negative charges. The molecule as a whole consists of no resultant charge. Molecules of this kind are termed as Non-polar. Illustrations of Non-polar molecules are air, hydrogen, carbon, benzene, tetrachloride and so forth. The second kind is the one in which the centre of positive charges and the centre of negative charges don't coincide. In this case, the molecule acquires a permanent dipole moment. This kind of molecule is termed as a Polar Molecule. Illustrations of polar molecules are water, glass and so on.

Therefore, we observe that, a Non-polar molecule obtains a Dipole Moment only in the presence of an electric field: while in a Polar Molecule the already existing dipole moment orients itself in the direction of the external electric field. Even in the polar molecules, there are a few induced dipole moments due to extra separation of charges, though this effect is comparatively much smaller than the reorientation effect and is therefore ignored for the polar molecules.

Polarization Vector 'P':

941_Polarization vector.jpg

Let us know the effect of an electric field on a dielectric material by keeping a dielectric slab among the two parallel plates as represented in the figure above. The electric field is set up through joining the plates to a battery.

In this, we restrict to a homogeneous and isotropic dielectric. A homogeneous and isotropic dielectric is one in which the electrical properties are similar at all points in all the directions. The applied electric field shifts the charge centers of the constituent molecules of the dielectric. The separation of the charge centers is shown the figure above. Here we discover that the negative charges of one molecule face the positive charges of its neighbor. Therefore in the dielectric body, the charges neutralize.

Though, the charges appearing on the surface of the dielectric are not neutralized. Such charges are termed as Polarization Surface Charges.

The whole effect of the polarization can be accounted for by the charges that appear on the ends of the specimen. The total surface charge, though, is bound and based on the relative displacement of the charges. This is reasonable to anticipate that the relative displacement of negative and positive charges is proportional to the average field 'E' within the specimen.

From the figure, we find out that such polarization charges appear only on such surfaces of the dielectric that are perpendicular to the direction of the field. No surface charges appear on the faces parallel to the field. Such a condition takes place merely in the special case of a rectangular block of dielectric kept among the plates of a parallel plate condenser.

The polarization of the material is quantitatively explained in terms of the dipole moment induced through the electric field. Keep in mind that the moment of a dipole comprising of charges q and -q separated through a displacement 'd' is given by P = - q d. It is recognized from experiments that the induced dipole moment (p) of the molecule raises with the raise in the average field 'E'. We can state that 'p' is proportional to 'E'.

Or p = α E

Here, α is the constant of proportionality termed as Molecular or Atomic Polarisability. Now we define a new vector quantity that we symbolize by 'P' and state it polarization of the dielectric or just polarization. Polarization 'P' is stated as the electric dipole moment per unit volume of the dielectric. This is significant to note that the word polarization is employed in a general sense to explain what happens in a dielectric if the dielectric is subjected to the external electric field. It is as well employed in the particular sense to indicate the dipole moment per unit volume.

Assume a 'n' polarized molecules each having a dipole moment 'p' present per unit volume of a dielectric and suppose all the dipole moments be parallel to one other. Then from the statement of P:

p = nP

From the definition above, units of P are:

Units of P = Coulomb m/m3 = Coulomb/m2 = C m-2

In common, 'P' is a point function based on the coordinates. In these cases, where the ideal condition mentioned above is not fulfilled, we would consider an infinitesimal volume 'V' all through which all the p's can be anticipated to be parallel and write the equation:

P = limΔV→0 i=1 N Pi/V   (Here, 'N' is the number of dipoles in volume 'V')

In this, 'V' is large compared to the molecular volume however small compared to ordinary volumes. Therefore, however p is a point function; it is a space average of p. The direction of p will, obviously, be parallel to the vector sum of the dipole moment of the molecules in 'V'.

Gauss' Law in a Dielectric:

We are familiar with the Gauss law in vacuum. In this, we modify and generalize it for dielectric material. Let assume two metallic plates as represented in the figure below. Let Eo be the electric field between such two plates.

871_Gauss law in dielectric.jpg

Now, we set up a dielectric material between the plates. If the dielectric is introduced, there is a drop in the electric field that implies a drop in the charge per unit area. As no charge consists of leaked off from the plates, such a reduction can be merely due to the induced charge appearing on the two surfaces of the dielectric. Due to this cause, the dielectric surface adjacent to the positive plate should encompass an induced negative charge and the surface adjacent to the negative plate should encompass an induced positive charge of equivalent magnitude. It is represented in the figure above.

The total charge within this volume is zero even although this material is polarized. The positive and negative charges are equivalent. For this volume the flux of field via the surface is zero. This can be written as:

surface at 1 E.dS = ∫s1 εo χ P. dS = 0

This represents that lines of 'P' are just similar to lines of 'E' apart from for a constant (εo). Rather than this, Gaussian volume, assume we take other one at area 2. In this Gaussian volume one surface is in the dielectric and the other is exterior to it. The curved surface is parallel to the lines of field (E or P). For this surface, the Gaussian volume outside the material 'P' is non-existent. Though, lines of 'P' should terminate within the Gaussian volume. Therefore the total flux of 'P' is finite and negative as the component of P normal to the surface, that is, n and σP the surface charge density are equivalent to one other in magnitude, then the surface integral be:

P.dS = Pn dS = - σp dS = -qp

Here qp is the charge in the Gaussian volume. Therefore, the flux of 'P' is equivalent to the negative of the charge included in the Gaussian volume.

Now, we generalize the Gauss' flux theorem. As the effects of polarized matter can be accounted for by the polarization surface charges, the electric field in any area can be associated to the sum of both free and polarization charges. Therefore in general:

closed surface E. dS = 1/εo (qf + qp)

Here qp stands for free charges and qp is the polarization charges.

Displacement Vector 'D':

This is one of the fundamental vectors for an electric field which mainly based on the magnitude of free charge and its distribution.

The electric displacement is stated by D = εo E + P; Gauss' law in dielectric is given by D. dS = qf dV. For an isolated charge 'q', kept at the centre of a dielectric sphere of radius 'r', we determine that the Gauss' flux theorem gives (that is, being a case of spherical symmetry)

(4πr2) (D) = q

That gives: D = qr/4πr2

∴ D = εE we get E = qr/4πεr2

From the above equation it follows that the force 'F', between the two charges q1 and q2, kept at a distance 'r' in a dielectric medium is represented by:

F = (q1q2/4πεr2) r

And the expression for the potential Φ at a distance r from q is: Φ = q/4πεr

Dielectric Strength and Breakdown:

We have observed that under the affect of an external electric field, polarization outcomes due to the displacement of the charge centers. In our explanation, we have treated the phenomenon as an elastic method. A question which occurs in our minds is, 'what would occur if the applied field is raised considerably?' One thing that is sure is that the charge centers will experience a significant pulling force. When the pulling force is less than the binding force between the charge centers, then the material will retain the dielectric property and on eradicating the field the charge centers will return to their equilibrium places. If the pulling force just balances the binding force, the charges will just be capable to overcome the strain of the separation and any slight imbalance will loosen the bonds among the electrons and the nucleus. A further raise of the applied field will outcome in the separation of the charges.

Once this occurs the electrons will be accelerated. The fast moving electrons will collide by the other atoms and multiply in number. This will outcome in the flow of conduction current. The minimum potential which causes the charge separation is termed as the breakdown potential and the procedure is termed as the dielectric breakdown.

Breakdown potential differs from substance to substance. It as well based on the thickness of the dielectric (that is, thickness measured all along the direction of the field). The field strength at which the dielectric is about to break down is termed as the Dielectric Strength. This is measured in kilovolts per metre. The knowledge of the breakdown potential is extremely significant for practical conditions, as in the use of capacitors in electrical circuits.

If a dielectric is subjected to a steadily increasing electric potential, a phase will be reached if the electron of the constituent molecule is torn away from the nucleus. Now the dielectric breaks down, that is, loses its dielectric properties and starts to conduct the electricity.

The breakdown voltage is the applied potential differences per unit thickness of the dielectric if the dielectric just breakdown.

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