## CHAPTER 6

## Dielectric Properties

#### 6.1 Introduction

Dielectrics are insulating materials. So, there are no free charge carriers in them. The dielectrics are of two types: (i) non-polar and (ii) polar dielectrics. In non-polar materials, the molecules are usually diatomic and composed of same type of atoms; each atom possesses a positive nucleus of charge *q* and surrounded by a symmetrically distributed negative electron cloud of charge −*q*. In the absence of an applied electric field, the centres of the positive and negative charge distribution coincide with each other. When external electric field is applied, then the centres of positive and negative charges move apart by a very small distance (10^{–10} m), then the molecules and atoms are said to be polarized. Next, the polar dielectric molecules are normally composed of two or more different types of atoms. They have dipole moments even in the absence of an external applied electric field. Usually, these molecular dipoles are oriented in random directions, so that the average dipole moment over the volume element is zero. In the presence of an externally applied electric field, the molecular dipoles tend to rotate certain extent in the direction of an applied electric field so that the material has some resultant dipole moment. Not only the rotation of molecular dipoles but also the centres of positive and negative charges of atoms are separated by small distance and it is called a dipole. It possess dipole moment (*p*). The electric dipole moment is defined as the product of one of the charge (*q*) and separation between the charges (d*l* ) [i.e., *p* = *q* d*l*]. The resultant dipole moment per unit volume of material is called polarization (*P*).

Dielectrics find applications in electrical and in electronic equipment. They are used for insulation purposes. In capacitors, dielectric material is used between the capacitor plates to increase capacitance.

#### 6.2 Dielectric constant

The permittivity of free space has been represented as *ε*_{0} and is equal to 8.85 × 10^{–12} F/m. The permitivity of any dielectric material can be represented as *ε* and it is equal to *ε _{r}*

*ε*

_{0}i.e.,

where *ε _{r}* is called the relative permittivity or dielectric constant of the dielectric material. It is a dimensionless quantity.

The dielectric constant can also be obtained from electric flux density (*D*) and an applied electric field (*E*). The number of electric force lines passing per unit area perpendicular to field is called electric flux density (*D*). It is proportional to the applied electric field (*E*).

If *P* is the polarisation of the dielectric material due to the applied electric field (*E*), Then the flux density ‘*D*’ is equal to flux density in vacuum plus polarisation of the material. Therefore, we have:

From Equations (6.2) and (6.3), we have:

The electric suscesptibility *χ* is:

From Equations (6.4) and (6.5), we have:

Experimentally, the dielectric constant can be obtained easily using parallel plate capacitor. If *C* and *C*′ are the capacitances of a capacitor without and with dielectric, respectively between the capacitor plates, Then, we have:

#### 6.3 Internal or local field

In dielectric solids, the atoms or molecules experience not only the external applied electric field but also the electric field produced by the dipoles. The resultant electric field acting on the atoms or molecules of dielectric substance is called the local field or an internal field.

To find an expression for local electric field on a dielectric molecule or an atom, we consider a dielectric material in the electric field of intensity *E*, between the capacitor plates so that the material is uniformly polarized, as a result opposite type of charges are induced on the surface of the dielectric near the capacitor plates. The local field is calculated by using the method suggested by Lorentz.

According to this method, consider a small spherical region of the dielectric with an atom at the centre of the sphere for which the local field is to be calculated. The radius of the sphere is chosen large enough so that the region outside the sphere is a continuum while inside the sphere as the actual structure of the substance. The part of the dielectric external to the sphere maybe represented by a system of charges induced and also at the spherical surface as shown in Fig. 6.1.

The electric field at the centre of the sphere may be written as:

**Figure 6.1** Local field

where *E*_{0} is the intensity of the electric field due to the charge ‘*q*’ on the plates, *E*_{p} is the field due to the polarization charges at the plate dielectric interface, *E*_{s} is the field due to the charges induced on the spherical surface and *E*_{m} due to all the dipoles of the atoms inside the spherical region. The macroscopic electric field (*E* ) inside the dielectric is:

For high symmetric crystals, *E*_{m} = 0. So, we write:

Equation (6.10) is not applicable to anisotropic materials. So, we consider isotropic materials only so that Equation (6.10) holds good. To evaluate *E*_{s}, an enlarged view of the spherical region in dielectric is shown in Fig. 6.2.

The charge element on a surface element d*s* of the sphere is equal to the parallel component of the polarization times the surface element i.e., *P* cos *θ*·d*s*.

Hence, the intensity of the electric field d*E*_{s} at the centre due to this charge element in the direction of ‘*r*’ is:

**Figure 6.2** Enlarged spherical region of dielectric

The components of d*E*_{s}, perpendicular to the direction of *P* will be cancelled due to an equal contribution from another symmetrically situated surface element. Only components of d*E*_{s} parallel to *P* will contribute to the integral of Equation (6.11) over the entire surface. Thus,

The surface element d*s* is the ring shown in Fig. 6.2 so that d*s* = *2πr* sin *θ* (*r* d*θ*) = *2πr*^{2} sin *θ* d*θ* and the limits of integration with respect to *θ* are from 0 to π. Thus, we have:

This can be evaluated making the substitution *Z* = cos *θ* and d*Z* = −sin *θ* d*θ*

So that,

Thus, equation (6.10) becomes:

Equation (6.14) is known as the Lorentz relation for local field.

#### 6.4 Clausius–Mosotti relation

Clausius–Mosotti relation makes relation between microscopic and macroscopic quantities of polarization. A dielectric material can be polarized by applying an external field on it. Dipole moment per unit volume of material is called polarization. The dipole moment ‘*p*’ is equal to the product of one of the charges and separation between the opposite charges of atoms or molecules. This dipole moment is proportional to the local electric field, *E*_{loc}, so that:

where ‘*α*’ is the electrical polarizability. If there are ‘*n*’ atoms per unit volume of the dielectric, then polarization ‘*P*’ is:

Substituting the equation for local electric field, *E*_{loc} = *E* + *P*/3*∈*_{0} in the above equation,

Polarization per unit electric flux density in vacuum is called electric susceptibility, represented as χ_{e}.

where *∈*_{0} is the permittivity of free space.

Substituting Equation (6.17) in (6.18) gives:

The expression for electric susceptibility can also be obtained from electric flux density. Let the number of electric force of lines per unit area of the surface perpendicular to the force of lines or the electric flux density ‘*D*’ is proportional to the intensity of the applied electric field.

The relation between the flux density and intensity of the applied electric field is:

where *∈* is the permittivity of the dielectric material and *∈*_{r} is its relative permittivity or dielectric constant of the material. The relation between *D* and *P* is:

Equations (6.20) and (6.21) are same:

From Equation (6.18), we have:

Substituting Equation (6.22) in (6.21) gives:

Equating Equations (6.20) and (6.23), we have:

Equations (6.19) and (6.24) are equal, so:

Adding 3 on both sides of Equation (6.25), we have:

Dividing Equation (6.25) by Equation (6.26) gives:

If there are *N* different types of atoms in the dielectric such that *n*_{1}, *n*_{2}, *n*_{3},…*n*_{N} are the number of first, second, third…*N* th kind of atoms and *α*_{1}, *α*_{2}, *α*_{3}…*α*_{N} are the polarizabilities of first, second, third…*N* th kind of atoms, respectively.

Then, Equation (6.27) becomes:

Here, and *α*_{i} is the polarizability of *i*th kind of atoms. If *ρ* is the density, *N*_{A} is Avogadro number and *M* is molecular weight of the crystal, then , so Equation (6.27) becomes:

Equation (6.28) or (6.29) is called Clausius-Mosotti equation. It can be used to determine the polarizabilities of the atoms if the dielectric constant is known. Further, the dielectric constants of new materials can be predicted from a knowledge of polarizabilities. This relation thus provides the necessary relation between the microscopic and macroscopic quantities of polarization.

#### 6.5 Orientational, ionic and electronic polarizations

When an electric field is applied on a dielectric crystal, then the positive charges of atoms and molecules are displaced along the field while the negative charges in a direction opposite to that of the applied field. This is the basis for polarization of a dielectric substance.

If a molecule has permanent dipole moment, then it is a dipolar molecule and the substance is a dipolar substance. Example is H_{2}O molecule. In the absence of an external electric field, the dipoles are randomly oriented, so that polarization is zero. But when the electric field is applied, these dipoles tend to rotate different extents in the direction of an applied electric field giving rise to dipolar or orientational polarization. The applied field also tends to displace the positive and negative ions of molecule in opposite directions causing a change in the ionic bond length. This change in bond length is to produce a net dipole moment in the crystal. This dipole moment per unit volume of material is known as ionic polarization.

The individual ions or atoms of a crystal are themselves polarized in the electric field. Fig. 6.3 shows the polarization of an atom, the electrons in its various shells are displaced relative to the nucleus and produce an electric dipole moment. This dipole moment per unit electric field of the material is called electric polarizability.

**Figure 6.3** (a) Unpolarized atom; (b) Polarized atom

The total polarizability (*α*) is the sum of the various polarizabilities such as the electronic polarizability (*α _{e}*), ionic polarizability (

*α*) and dipolar polarizability (

_{i}*α*). Therefore, we write

_{d}*α*=

*α*+

_{e}*α*+

_{i}*α*. Now, we study each polarization in detail.

_{d}#### (a) Dipolar or orientational polarization

The expression for dipolar polarization can be obtained from Langevin-Debye theory as given below. According to Debye, oriental polarization is due to the rotation of polar molecules in dielectric substance. In the absence of an applied electric field, the dipoles of the substance are randomly oriented in all directions with equal probability and the resultant polarization is zero. In the presence of an applied electric field (*E* ), the torque (*τ*) acting on a dipole to rotate it in the direction of *E* is given as:

where *p* is the dipole moment of a molecule [Fig. 6.4]. The only force that prevents permanent dipoles from complete alignment with the field is thermal agitation.

Therefore, an equilibrium state will reach in which different dipoles will make 0 to π radian angles with field direction, producing a net resultant polarization in the direction of the field. The potential energy (*V*) of a dipole corresponding to an angle *θ* between *p* and *E* direction is:

**Figure 6.4** Torque acting on a dipole

According to Boltzmann distribution law, the probability for a dipole to make an angle between *θ* and *θ* + d*θ* with the field is proportional to:

where 2π sin*θ* d*θ* is the solid angle between *θ* and *θ* + d*θ*. Hence, the average component of the dipole moment along the field direction is equal to:

*θ* = 0 corresponds to parallel alignment and *θ* = *π* to anti-parallel alignment of the dipoles. Dividing numerator and denominator by 2*π* and putting cos *θ* and d*x* = −*a* sin *θ* d*θ*.

Substituting the above values in Equation (6.31), we have:

*L*(*a*) is called Langevin function, because this formula was derived by Langevin in 1905 in connection with paramagnetism.

A graph of *L*(*a*) versus ‘*a*’ has been plotted as shown in Fig. 6.5. Near the origin, the Langevin function increases linearly so that *L*(*a*) = *a*/3. As ‘*a*’ increases, the function continues to increase and approaching the saturation value unity as *a* → ∞ i.e., for high field strengths. This saturation corresponds to complete alignment of the dipoles in the field direction, so that <cos *θ*> = 1

**Figure 6.5** A graph plotted between *L*(*a*) and ‘*a*’

For moderate field strengths (when *a* << 1), *L*(*a*) = *a*/3

If there are *N* molecules per unit volume of the crystal, then the dipolar polarization (*P _{d}*) is:

The average dipolar dipole moment [ *p* <cos *θ*> ] is proportional to the intensity of the applied electric field [*E*] i.e., *p* <cos *θ**>* ∝ *E* (or) *p* <cos *θ*> = *α*_{d} *E*, where the proportionality constant *α*_{d} is called the dipolar polarizability given as:

Equation (6.34) is actually applicable to liquids and gases, because in these substances only the molecular dipoles may rotate continuously and freely, as has been assumed in its derivation. In solids, a dipole may move back and forth between certain limits, which depends on the temperature and electric field. Therefore, the total polarizability of a dipolar molecule can be written as:

where *α _{ei}* is the combined polarizability from electronic and ionic contributions. The Clausius–Mosotti equation for a dipolar system is:

Equation (6.36) is known as Debye formula. A plot can be drawn between versus 1/*T* as shown in Fig. 6.6

The graph is a straight line, the slope of this line is proportional to *p*^{2} and its intercept is proportional to *α _{ei}*. This formula leads to the determination of both the dipole moment and

*α*.

_{ei}

**Figure 6.6** Temperature versus dielectric constant

#### (b) Ionic polarization

Polarization in ionic crystals arises due to the displacement of ions from their equilibrium positions by the force of an applied electric field.

Ionic polarization can be calculated by considering NaCl crystal. Let the masses of Na^{+} and Cl^{−} ions are *m* and *M*, respectively. In the absence of an applied electric field, the Na^{+} and Cl^{−} ions are at equilibrium positions and the equilibrium separation between these ions is equal to *r*_{0} (say). After application of the electric field of intensity *E*, on the NaCl crystal, some amount of force equal to *eE* acts on each Na^{+} and Cl^{−} ions in opposite directions, so that the ions get displaced by *x*_{1} and *x*_{2} distances from equilibrium position. These displacements of ions produce dipole moment in the molecules. The induced dipole moment (*p*) per molecule is:

Even though the electric field is continuously acting on the ions, the displacement between the ions will not continuously increase because of the restoring force between the oppositely charged ions. At equilibrium conditions the restoring force (*F*) between the ions is:

*F* = *K*_{1}*x*_{1} = *K*_{2} *x*_{2} where *K*_{1} and *K*_{2} are force constants

From the above equation, for Na^{+} ion of mass ‘*m*’,

where *ω*_{0} is the natural frequency of NaCl molecule. For Cl^{−} ion of mass ‘*M*’,

From Equations (6.38) and (6.39), we have

Substituting Equation (6.40) in (6.37) gives:

This is the induced dipole moment in NaCl molecule. If *N* number of NaCl molecules are present per unit volume of the crystal, then polarization *P* is:

The ionic polarizability, *α _{i}* is

Substituting the various values in the above equation, we get *α*_{i} = 9.75 × 10^{−24} F–m^{2}, but the experimental value is 3.3 × 10^{−24} F-m^{2}. The poor agreement between these values is that the effective ionic charge on an ion was assumed as *e*, but in fact it turned out to be 0.7*e*.

#### (c) Electronic polarization

Electronic polarization can be calculated by considering the atoms of a given substance. In the absence of an applied electric field, an atom will be spherical as shown in Fig. 6.6(a). The atom consists of a point nucleus of charge +*Ze*, surrounded symmetrically by an electron cloud of charge −*Ze* in a sphere of radius *r*. If an electric field *E* is applied on the atom, then a force of |*ZeE*| acts on the nucleus in the direction of the applied electric field and on the electron cloud in the opposite direction, so that they shift with respect to each other by a distance ‘*d*’ as shown in Fig. 6.6(b). The electron cloud is assumed to remain spherical for simplicity.

**Figure 6.6** Electric polarization

The distance of separation ‘*d*’ between the centre of electron cloud and nucleus is such that the restoring force on the nucleus and electron cloud is equal to the force of attraction between the nucleus and the fraction of the charge inside the sphere of radius ‘*d*’. Applying Coulomb's law for restoring force,

This is equal to force by electric field,

Equating these forces, we have:

The induced dipole moment ‘*p _{e}*’ is:

The dipole moment per unit volume is polarization.

If ‘*N*’ number of atoms are present in unit volume of material, then electronic polarization *P _{e}* is:

and the polarizability *α _{e}* is:

Using Equation (6.49), electronic polarizability can be calculated.

For monoatomic gas,

Hence, we have:

*ε*_{r} = 1 + 4πr ^{3}*N*

The value 4π*r*^{3} *N* is of the order of 10^{–4}. Hence, *ε _{r}* ≈ 1 for gases. In solids,

*ε*varies from 2 to 10.

_{r}#### 6.6 Frequency dependence of polarizability: (Dielectrics in alternating fields)

In this topic, the variation of polarizability, polarization and dielectric constant of the dielectric with the frequency of the applied electric field has been explained. The permittivity of a dielectric material is equal to *ε*_{0}*ε*_{r}, where *ε*_{r} is called the relative permittivity or dielectric constant of the material and *ε*_{0} is the permittivity of the free space. Also from Clausius–Mosotti relation, we know that the dielectric constant is related to polarizability of the material. So, we can see the variation of permittivity and hence relative permittivity or dielectric constant, in turn polarizability of a dielectric material with the frequency of the applied electric field. When an alternating electric field of frequency less than 10^{6} Hz is applied on a dielectric material, then the orientation of the electric dipoles and hence polarization will reverse every time as the polarity of the field reverses. The polarization of the material follows the field without any lag so that the permittivity remains constant. As the frequency of the applied electric field is increased from 10^{6} Hz to 10^{11} Hz (radiowave frequencies), the electric dipoles present in the material unable to follow the field, hence they lag behind the field and orientational polarization ceases. So, the dielectric constant changes whereas ionic and electronic polarizations are present. Again if the frequency of the applied electric field is increased from 10^{11} Hz to 10^{14} Hz [infrared frequencies], the heavy positive and negative ions present in the material cannot follow the field variations; hence ionic polarization ceases.

This leads againt to the change in dielectric constant. The electronic polarization exists up to a frequency of nearly 10^{15} Hz, because electrons are light particles and easily follow the variations of the applied voltage.

To know the dependence of electronic polarizability with the frequency of the applied electric field in the optical region, we consider an atomic model with a nucleus of charge +*e* and an electron. The electron has been represented as a cloud having radius *r*_{0}. In the absence of an applied electric field, the centre of electron cloud coincides with the nucleus. After applying static electric field, the centre of electron cloud displaces by small distance *x* (say) relative to the nucleus.

Then, the restoring force

where = restoring force constant.

If there is no damping, then the equation of motion is:

or

The solution for Equation (6.52) is of the form *x* = *x*_{0} sin (*ω*_{0}*t* + *δ*), where *x*_{0} = maximum displacement, *δ* is the integrating constant and is the natural or resonance frequency of the electron cloud. If we consider damping in the motion of electrons, then Equation (6.51) becomes:

where 2*b* is the damping constant.

Instead of static electric field, an alternating electric field *E* = *E*_{0} cos*ωt* is applied on the electron, then the Lorentz force, –*eE*_{0} cos*ωt* acts on the electron cloud and the equation of motion is:

where *E*_{0} is the maximum electric field.

(or)

The solution for Equation (6.55) is of the form:

where *A** is complex amplitude

Here, Real [*E*_{0}*e ^{iωt}*] =

*E*

_{0}cos

*ωt*

Equation (6.56) is differentiated twice and substituted in Equation (6.55)

Substituting these in Equation (6.55)

As the exponential value in the above equation is not equal to zero, so the value in curling braket is equal to zero.

we have:

Substitute Equation (6.58) in Equation (6.56)

The induced dipole moment *p*_{ind} (*t*) = –*ex* (*t*)

Under static electric field, the electronic induced dipole moment (*p*_{ind}) is proportional to the applied electric field *E*,

So,

where *α _{e}* is electronic polarizability.

Comparing Equations (6.60) and (6.61), the coefficient of electric field *E*_{0} *e* ^{iωt} is the electronic polarizability. So,

where *α*_{e}* is the complex electronic polarizability and is equal to:

To separate real and imaginary parts of Equation (6.62), multiply and divide with

We have:

The above equation can be represented in the form as

Here, *α′ _{e}* and

*α″*are the real and imaginary parts of polarizability, respectively. The induced electronic dipole moment per unit volume of the material is the electronic polarization of the material. This can be represented as:

_{e}

where *N* is the number of atoms per unit volume of the material

Substituting *α′ _{e}* and

*α″*in Equation (6.66), we have:

_{e}

The first part of polarization is in phase with the applied electric field, whereas the second part of polarization lags 90° with the applied field.

Equation (6.65) can also be represented in terms of dielectric constant as:

So,

Substitute *α*_{e}* from Equation (6.63) in the above Equation 6.68, we have:

or

A graph has been plotted for the real and imaginary parts of *α*_{e}* (Equation (6.63)) with *ω* as shown in Fig. 6.7.

In Equation (6.62) for *ω* = 0 (i.e., dc field), the imaginary part becomes zero and the real part is:

For *ω* < *ω*_{0}, the real and imaginary parts of electronic polarizabilities are positive.

For *ω* > *ω*_{0}, the real part is negative and the imaginary part is positive.

**Figure 6.7** Variation of *α*′_{e} and *α*″_{e} with *ω* for a single electron

At *ω* = *ω*_{0}, the real part is zero and the imaginary part has maximum value. From the graph, it has been observed that the real part of polarizability is almost constant up to large frequencies from zero frequency, whereas the imaginary part vanishes at *ω* = 0 and when *ω* → ∞.

So far, we have discussed with an atomic model in which there is only one electron in the atom. Actually, majority of atoms have several electrons and each electron has its own restoring force constant (*f*) and damping constant (*b*). In multielectron atoms, there are several values of *ω*_{0} (natural frequencies of different atoms). For multielectron atom, plots have been drawn for *α*′_{e} and *α*″_{e} versus *ω* and are shown in Fig. 6.8.

**Figure 6.8** A graph has been drawn between, *α*′_{e}, *α*″_{e}, versus *ω* for multielectron atom

#### 6.7 Piezoelectricity

The word ‘Piezo’ means ‘pressure’ in Greek. So, ‘piezoelectricity’ means ‘pressure electricity’. Piezoelectric phenomenon was discovered by Curie brothers in 1880. Piezoelectric effect is shown by certain non-centrosymmetric crystals, such as quartz, rochelle salt, tourmaline and barium titanate. Electric polarization develops opposite charges on their surfaces by stress. On these substances, a mechanical stress produces an electric polarization and reciprocally, an applied electric field produces a mechanical strain. These effects are called the direct and inverse piezoelectric effects. Crystals with centres of inversion do not exhibit piezoelectricity.

A crystal can exhibit piezoelectricity only if its unit cell lacks centre of inversion. Fig. 6.9(a) shows the three-fold symmetry axis of an unstressed quartz crystal. The arrows represents the dipole moments. The sum of three dipole moments at the vertex is zero. When subjected to stress, it gets polarization (*P*) in the direction indicated because of the distortion of the charge symmetry.

**Figure 6.9** Three-fold symmetry of quartz crystal: (a) when it is unstressed and (b) when it is stressed

Piezoelectric effect in quartz crystal is explained here. Quartz crystallizes in hexagonal crystal system. A section cut perpendicular to Z-axis (optic axis) is shown in Fig. 6.10. Optic axis in a crystal is a direction along which the velocity of ordinary and extraordinary rays is the same. The lines joining the opposite corners are called X-axis and the lines perpendicular to the opposite faces are known as Y-axis. A plate of quartz crystal with its faces perpendicular to X-axis is called X-cut quartz crystal. Similarly, a plate with its faces perpendicular to the Y-axis is called Y-cut crystal. Quartz crystal will not show piezoelectric effect along the optic axis, piezoelectricity is maximum along Y-axis and is medium along X-axis.

**Figure 6.10** Quartz crystal: Section cut perpendicular to Z-axis

**Applications:**

- Piezoelectric effect is used in detection and to produce sound waves.
- Quartz crystal responds to pressure variations; so, it is used as a pressure transducer.
- The natural frequency of quartz crystal does not vary with temperature. Using this property, quartz crystal is used to produce highly stable RF oscillations for broadcasting purposes and in quartz watches to maintain accurate time.

#### 6.8 Ferroelectricity

Few dielectric substances such as barium titanate [BaTiO_{3}], rochelle salt, KDP (KH_{2}PO_{4}), ADP [NH_{4}H_{2}PO_{4}], LiNbO_{3}, KNbO_{3}, etc. called ferroelectric materials show spontaneous electric polarization (*P*) below Curie temperature (*T* < *T*_{c}) is known as ferroelectricity. This is shown in Fig. 6.11(a). Polarization of a material without any applied external electric field on it below Curie temperature is known as spontaneous polarization.

The dielectric constant of a ferroelectric material increases enormously as the temperature of the material reduces to its Curie temperature. The variation of dielectric constant with temperature is shown in Fig. 6.11(b) and is given by Curie-Weiss law where *C* is the Curie constant and *T*_{c} is Curie temperature.

Another important property of ferroelectrics is that they show hysteresis similar to magnetic materials under the action of alternating voltages as shown in Fig. 6.11(c). When an electric field is applied on the specimen, the polarization increases along the curve OABC. And when the field is reduced to zero, then a certain amount of polarization called remanent polarization, *P*_{r} is still present in the material. To remove this polarization, electric field in the opposite direction must be applied. The amount of field required to remove remanent polarization is called coercive field (*E*_{c}).

**Figure 6.11** (a) Spontaneous polarization; (b) Variation of dielectric constant with temperature; (c) Hysteresis of a ferroelectric material

The spontaneous polarization of ferroelectric material is due to asymmetrical ionic displacements in the crystal structure as it is cooled through *T*_{c}. This can be explained by considering BaTiO_{3} as an example. The structure of BaTiO_{3} above *T*_{c} (=120° C) is shown in Fig. 6.12. The unit cell of BaTiO_{3} above Curie temperature is cubic. Oxygen ions are present at the centres of six cube faces. These six oxygen ions form an octahedron configuration.

**Figure 6.12** Ba TiO_{4} unit cell

The Ti^{4+} ion is present at the centre of this octahedron. Barium ions occupy the corners of the cube. The centre of the negative charges coincide with that of the centre of positive charges. So, the net dipole moment is zero. As it is cooled through *T*_{c}, the Ti^{4+} and Ba^{2+} ions move with respect to O^{2−} ions. X-ray and neutron diffraction studies show that the titanium and barium ions move up by 2.8% and oxygen ions move down by 1%. This favours spontaneous polarization of the material. The direction of spontaneous polarization may lie along any one of the cube edges, giving six possible directions of polarization; the material expands in the direction of polarization and contracts perpendicular to that direction. Thus, the unit cell changes from cubic to tetragonal structure. Now, the centre of positive charges is no longer coincident with the centre of negative charges. This explains spontaneous polarization.

**Applications:**

- The ferroelectric material possesses very high value of dielectric constant, so they are used in the manufacture of small-sized and large-capacitance capacitors.
- Because of the hysteresis property of ferroelectric materials, they are used in the construction of memory devices used in computers.
- The ferroelectric materials show piezoelectric property, so they are used to produce and detect sound waves.
- The ferroelectric materials also show pyroelectric property, so they are used to detect infrared radiation.

#### 6.9 Frequency dependence of dielectric constant

The permittivity (*∈*) of a dielectric material is equal to *∈*_{0} *∈*_{r}. Where *∈*_{r} is called relative permittivity or dielectric constant of the material and *∈*_{0} is the permittivity of free space. Also from Clausius-Mosotti relation, we know that the dielectric constant is related to polarizability of the material. So, we can see the variation of permittivity and hence relative permittivity or, dielectric constant, in turn, polarizability of a dielectric material with the frequency of the applied electric field. The fall in the permittivity of a dielectric material with increasing frequency of the applied electric field is usually referred to as anomalous dispersion.

The behaviour of a dielectric material in alternating electric field shows that the dielectric constant is a complex quantity. The imaginary part of this dielectric constant represents the dielectric loss of the material. When an alternating electric field of frequency less than 10^{6} Hz is applied on a dielectric material, then the orientation of the electric dipoles and hence the polarization will reverse every time as the polarity of the field reverses. The polarization of the material follows the field without any lag so that the permittivity remains constant. As the frequency of the applied electric field is increased from 10^{6} Hz to 10^{11} Hz [radiowave frequencies], the electric dipoles present in the material are unable to follow the field, hence they lag behind the field and orientational polarization ceases. So, the dielectric constant changes whereas ionic and electronic polarizations are present. Dispersion arising during the transition from full orientational polarization at zero or low frequencies to negligible orientational polarization at high radio frequencies is referred to as dielectric relaxation. Again, if the frequency of the applied electric field is increased from 10^{11} Hz to 10^{14} Hz [infrared wave frequencies], the heavy positive and negative ions present in the material cannot follow the field variations, hence ionic polarization ceases. This leads again to the change in dielectric constant. The dispersion arising during the transition from full atomic polarization at radio-frequencies to negligible atomic polarization at optical frequencies is referred to as resonance absorption. The electronic polarization exists up to a frequency of nearly 10^{15} Hz, because electrons are light particles and easily follow the variations of applied voltage. Above this frequency, all polarizations ceases.

Now, we will see the variation of the real part and the imaginary part of dielectric constant in orientational, ionic and electronic polarizations.

#### Orientational polarization

The complex dielectric constant can be expressed as:

As shown in Fig. 6.13(a), the real part of dielectric constant *∈*′_{e} (*ω*) is constant equal to *∈*(0) for all frequencies in the range where *τ* is the orientational relaxation time and *ω* is the frequency of the applied voltage. This frequency range usually covers all frequencies up to the microwave region. In the frequency range *ω* ≥ 1/*τ*, the real part of dielectric constant decreases to a constant value equal to *n _{2}* [called optical dielectric constant], where

*n*= refractive index of the material. The imaginary part,

*∈*″(

*ω*) has its maximum value at

*ω*= 1/

*τ*[or

*ωτ*= 1] and decreases as the frequency departs from this value (i.e., increases or decreases), and represents dissipation of electrical energy in the form of heat in the dielectric material. The rate of dissipation is proportional to

*∈*″(

*ω*) and is maximum at

*ω*= 1/

*τ*.

#### Ionic polarization

In the high frequency range (IR), the ionic contribution vanishes because at high frequencies, the ions cannot follow the oscillations of the field.

#### Electronic polarization

The real part of the dielectric constant *∈*′(*ω*) gives the value of the dielectric constant and *∈*″(*ω*) gives the power dissipated and hence the damping loss, the variations of these are shown in Fig. 6.13(b). *∈*″(*ω*) has a maximum at *ω* = *ω*_{0}, this means that the material absorbs energy at the natural frequency. This is called resonance absorption. *∈*′(*ω*) is strongly frequency-dependent and the susceptibility undergoes a change in sign is called anomalous dispersion.

Since the dielectric constant of a material is related with the polarizability ‘*α*’ of a dielectric substance. The variation of polarizability with frequency is shown in Fig. 6.14.

#### 6.10 Important requirements of insulators

Insulating materials with different properties are required in electrical, thermal, mechanical and chemical applications.

**Figure 6.13** Real and imaginary parts of dielectric constant with frequency of alternating voltage: (a) Upto microwave region and (b) At optical frequencies

**Figure 6.14** Variation of polarizability with frequency of alternating voltage

#### (a) Electrical requirements

High-resistivity insulating materials are required to reduce leakage current. To withstand at high voltages, high dielectric strength materials are used. Dielectric strength is defined as the ability of a material to withstand up to a certain maximum electric field without breakdown. An insulator should have, low loss factor, high thermal stability, good mechanical strength, high resistivity and high dielectric strength. Mostly, polymeric materials like polystyrene, polyethylene, polyvinyl chloride, acrylic plastic, kapron, etc. are used.

Insulating materials used in capacitors should have high permittivity, low loss factors, high resistivity values, low frequency dependence of loss factor, good thermal stability, high dielectric strength and dielectric constant in the frequency range of operation. Good capacitor materials are TiO_{2}, SnO_{2} (or) ZrO_{2}, CaO, MgO or their mixtures.

Phenolics are widely used as insulating varnishes. Laminated sheets are used as insulating components of generators, transformers, etc. Teflon is one of the best insulators, as it has high resistivity.

#### (b) Thermal requirements

Some insulating materials such as transformer oil, hydrogen, helium, etc. are used for insulation and cooling purposes. Good thermal conductivity is desired for the materials which are used as coolants. The insulators should possess small coefficients of thermal expansion to prevent mechanical damage. The insulators should be non-ignitable, if ignitable it should be self-extinguishable.

#### (c) Mechanical requirements

Depending on the use, the insulating materials should have some required mechanical properties. For example, when an insulator is used in electric machine, it should have sufficient mechanical strength to withstand vibrations and shock. Insulators are used on the basis of volume and not on weight, hence a low-density insulator is preferred.

#### (d) Chemical requirements

Chemically insulating materials should be resistant to oils, acids, alkalies, gas fumes and liquids. Insulators should be non-absorbant of water, because by absorbing water the insulating resistance and dielectric strength of the material is reduced.

#### Formulae

#### Solved Problems

*1. A solid elemental dielectric with 3×10 ^{28}* atoms/m

^{3}

*shows an electronic polarizability of 10*F-m

^{−40}^{2}

*. Assuming the internal electric field to be a Lorentz field, calculate the dielectric constant of the material.*

(Set-3–Sept. 2007), (Set-1–May 2004), (Set-4–Nov. 2004), (Set-1–May 2003)

**Sol:** Number density of dielectric atoms, N = 3 × 10^{28}/m^{3}

Electronic polarizability, *α _{e}* = 10

^{−40}F-m

^{2}

Calculate the dielectric constant, *∈ _{r}* = ?

*2. A parallel plate capacitor has an area of 100* cm^{2}*, a plate separation of 1* cm *and is charged to a potential of 100* V*. Calculate the capacitance of the capacitor and the charge on the plates.*

(Set-4–May 2007), (Set-4–May 2004), (Set-3–Nov. 2004), (Set-4–May 2003)

**Sol:** Area of the capacitor plates, *A* = 100 cm^{2} = 10^{−2} m^{2}

Separation between the plates, *d =* 1 cm = 10^{−2} m

Potential between the plates, *V =* 100 V

Capacitance, *C =* ? and charge on plates, *Q* = ?

We know that:

*3. The dielectric constant of He gas at NTP is 1.0000684. Calculate the electronic polarizability of He atoms if the gas contains 2.7 ×10 ^{25}* atoms per m

^{3}

*.*

(Set-1–Sept. 2007), (Set-4–June 2005), (Set-2–May 2004), (Set-2–May 2003), (Set-4–Nov. 2003)

**Sol:** The dielectric constant, *∈ _{r}* = 1.0000684

Number density of He atoms, *N =* 2.7 × 10^{25}/m^{3}

Electronic polarizability, *α _{e}* = ?

*4. A parallel plate of area 650* mm^{2} *and a plate separation of 4* mm *has a charge of 2 × 10 ^{−10}* C

*on it. When a material of dielectric constant 3.5 is introduced between the plates. What is the resultant voltage across the capacitor?*

(Set-1–May 2007), (Set-1–Nov. 2003)

**Sol:** Area of the capacitor plates, *A* = 650 mm^{2} = 650 × 10^{−6} m^{2}

Distance of separation between the plates, *d* = 4 mm = 4 × 10^{−3} m

Charge on the plates, *Q* = 2 × 10^{−10}C

Dielectric constant of the material between the plates, *∈ _{r}* = 3.5

Voltage across the capacitor, *V =* ?

*5. A parallel plate capacitor having an area 6.45 × 10 ^{−4}* m

^{2}

*and a plate separation of 2 × 10*

^{3}m, across which a potential of 12 V is applied. If a material having a dielectric constant 5.0 is positioned within the region between the plates, compute the polarization.

(Set-3, Set-4–May 2006), (Set-2–Nov. 2003)

**Sol:** Area of the plates, *A* = 6.45 × 10^{−4} m^{2}

Separation between the plates, *d = 2* × 10^{−3} m

Potential across the plates, *V =* 12 V

Dielectric constant, *∈ _{r}* = 5

Polarization, *P* = ?

Intensity of the electric field,

*6. The relative dielectric constant of sulphur is 3.75 when measured at 27 °* C. *Assuming the internal field constant γ= 1/3, calculate the electronic polarizability of sulphur if its density at this temperature is 2050* Kg/m^{3}. *The atomic weight of sulphur being 32.*

(Set-1–June 2005), (Set-3–May 2004), (Set-3–May 2003)

**Sol:** The dielectric constant of sulphur, *∈ _{r}* = 3.75

Internal field constant,

Density of sulphur, *D* = 2050 Kg/m^{3}

Atomic weight of sulphur, *M =* 32

Electronic polarization *α _{r}* = ?

Number of atoms/m^{3}, where *N _{A}* = Avogadro number

Hence, from Clausius–Mosotti relation

*7. There are 1.6 × 10 ^{20} NaCl* molecules/m

^{3}

*in a vapour. Determine the orientational polarization at room temperature if the vapour is subjected to a dielectric field 5000*V/Cm

*. Assume that the NaCl molecule consists of sodium and chlorine ions separated by 0.25 nm.*

(Set-2-June 2005)

**Sol:** Number of NaCl molecules, *N =* 1.6 **x** 10^{20}/m^{3}

Room temperature, *T* = 300 K

Intensity of electric field, *E* = 5000 V/Cm = 5 **x** 10^{5} V/m

Separation between ions, *x =* 0.25 nm = 0.25 × 10^{−9} m

Orientation polarization, *P _{d}* = ?

dipolemoment, *p = ex*

where *K _{B}* = Boltzmann constant = 1.381 × 10

^{−23}J/K

*8. The dielectric constant of helium, measured at 0°* C *and 1 atmosphere is ∈_{r} = 1.0000684 Under these conditions the gas contains 2.7 × 10 ^{25}* atoms/m

^{3}

*. Calculate the radius of the electron cloud. Also calculate the displacement when a helium atom is subjected to an electric field of 10*V/m.

^{6}

(Set-3–Sept. 2006)

**Sol:** The dielectric constant, *∈ _{r} = 1.0000684*

Number density of He atoms, *N =* 2.7 × 10^{25} atoms/m^{3}

Electronic polarizability, *α _{e}* = ?

Electric field applied, *E* = 10^{6} V/m

Radius of electron cloud, *r* = ?

Displacement of centres of charges, *d* = ?

*9. A parallel plate capacitor of area 750* mm^{2} *possess a charge of 2.5 × 10 ^{−10}* C

*when the plates are separated by 5*mm

*and the space between plates is filled with a material of dielectric constant of 3.5. Find the voltage across the capacitor plates.*

**Sol:** Area of plates, *A* = 750 mm^{2} = 750 × 10^{−6} m^{2}

Separation of plates, *d* = 5 mm = 5 × 10^{−3} m

Charge on plates, *Q* = 2.5 × 10^{−10}C

Dielectric constant of material, *∈ _{r}* = 3.5

Voltage across the plates, *V = ?*

Equation (1) = (2)

*10. A monoatomic gas contains 3 × 10 ^{25}* atoms/m

^{3}

*at 1 atmospheric pressure and at room temperature. The radius of gaseous atoms is 0.2*nm

*. Find the dipole moment per unit electric field, polarization, dielectric constant and polarizability.*

**Sol:** Number of atoms per unit volume, *N =* 3 × 10 ^{25}/m^{3}

Radius of atoms, *r =* 0.2 nm = 0.2 × 10^{−9} m

Dipole moment *p* = ?

Polarization, *P = ?*

Dielectric constant, *∈ _{r}* = ?

and Polarizability, *α _{e}* = ?

Dipole moment per unit electric field, *p* = 4*π∈ _{0}r^{3}*

Polarization, *P = Np* = 3 × 10 ^{25} × 8.9 × 10 ^{−40} = 26.7 × 10 ^{−15} C-m

To find dielectric constant, *∈ _{r}*:

*11. The relative permittivity of argon at 0 ° C and at 1 atmospheric pressure is 1.000435. Calculate the polarizability of the atom if the gas contains 2.7 x 10 ^{25}* atoms/m

^{3}

*. Given*F/m

*∈*= 8.85 × 10_{r}^{−12}**Sol:** Relative permittivity, *∈ _{r}* = 1.000435

Number density of atoms, *N =* 2.7 × 10^{25} atoms/m^{3}

Polarizability of the atom, *α _{e}* = ?

*12. If the relative permittivity of sulphur is 4.0, calculate its atomic polarizability. [given that sulphur in cubic form has a density of 2.08 × 10 ^{3} kg/m^{3} and its atomic weight is 32]*

**Sol:** Relative permittivity of sulphur, *∈ _{r}* = 4.0

Polarizability, *α _{e}* = ?

where N = number density of atoms

#### Multiple Choice Questions

- Dielectrics are: ( )
- metals
- semiconductors
- insulating materials
- none

- Local electric field is calculated by using the method suggested by: ( )
- Lorentz
- Weiss
- Curie
- Coulomb

- If
*P*is the polarization of a dielectric material of dielectric constant*∈*and_{0}*E*is the macroscopic electric field, then internal field is: ( ) - A dielectric material can be polarized by applying________field on it. ( )
- magnetic
- gravitational
- electric
- meson

- Polarization per unit applied electric field is called: ( )
- electric susceptibility
- magnetic susceptibility
- electric polarization
- dielectric constant

- In the absence of an external electric field on a dipolar substance, the electric dipoles are: ( )
- parallel
- alternatively anti-parallel
- randomly oriented
- none

- The total polarizability of a substance is equal to: ( )
- orientational and ionic polarizabilities
- ionic and electronic polarizabilities
- both a and b
- none

- The dipolar polarizability for low applied electric fields, at temperature ‘
*T*’ on a dipolar substance is [*K*_{B}=Boltzmann constant and*p*= dipole moment] ( ) - Dipolar polarization is actually applicable to: ( )
- gases
- liquids
- solids
- both a and b

- The observed ionic polarizability of NaCl molecule is: ( )
- 1.3 × 10
^{−24}F-m^{2} - 3.3 × 10
^{−24}F-m^{2} - 3.1 × 10
^{−24}F-m^{2} - 5.3 × 10
^{−24}F-m^{2}

- 1.3 × 10
- The effective ionic charge in NaCl crystal is: ( )
- 1
*e* - 1.3
*e* - 0.7
*e* - 0.5
*e*

- 1
- If
*r*is the radius of an atom and*∈*is the permittivity of free space, then electronic polarizability is: ( )_{0}- 4
*π∈*_{0}r^{2} - 4
*π∈*_{0}r^{3} - 4
*π*^{2}∈_{0}r^{3} - 4
*π∈*_{0}^{2}r^{3}

- 4
- Piezoelectric effect is shown by: ( )
- quartz
- rochelle salt
- barium titanate
- all the above

- Piezoelectric effect in quartz crystal is maximum along: ( )
- X-axis
- Y-axis
- Z-axis
- optic axis

- Piezoelectric effect is used: ( )
- to produce sound waves
- to detect sound waves
- as a pressure transducer
- all

- Quartz crystal is used: ( )
- to produce highly stable RF oscillations for broadcasting
- in watches to maintain accurate time
- both a and b
- none

- Ferroelectric materials are used: ( )
- to detect infrared radiation
- to produce and detect sound waves
- in the construction of memory devices
- all

- Ferroelectric materials are: ( )
- barium titanate and rochelle salt
- KH
_{2}PO_{4}and NH_{4}H_{2}PO_{4} - LiNbO
_{3}and KNbO_{3} - all the above

- The ionic polarization ceases at_________ frequency of the applied electric field. ( )
- 10
^{11}Hz - 10
^{13}Hz - 10
^{14}Hz - 10
^{6}Hz

- 10
- Insulating material used in capacitors should have: ( )
- high permittivity and low loss factors
- high resistivity and low frequency dependence of loss
- good thermal stability and high dielectric strength
- all

- Chemically, an insulating material should be resistant to: ( )
- oils and acids
- alkalies and gas fumes
- liquids
- all the above

- Dielectric material is used between the capacitor plates to: ( )
- increase electric field
- increase capacitance
- decrease capacitance
- decrease electric field

- Dipole moment is defined as the ________of one of the charge and separation between the charges. ( )
- product
- sum
- ratio
- none

- The resultant electric field acting on the atoms or molecules of dielectric substance is known as ________. ( )
- local field
- internal field
- both a & b
- none

- Clausius-Mosotti relation makes relation between microscopic and macroscopic quantities of: ( )
- electric field
- capacitance
- polarization
- none

- Dipole moment per unit volume of material is called: ( )
- polarization
- polarizability
- both a & b
- none

- In the absence of an applied electric field on a dipolar substance, the polarization is: ( )
- finite
- zero
- high
- all the above

- By applying electric field on a dipolar substance, it results in _________ polarization. ( )
- electrical
- ionic
- orientational
- all the above

- The total polarizability of a substance, it is the sum of _________ polarizabilities: ( )
- dipolar
- ionic
- electric
- all the above

- Orientational polarization is due to the_______of polar molecules in dielectric substance. ( )
- rotation
- change in separation
- both
- none

- In Greek, piezoelectricity means: ( )
- pressure electricity
- thermal electricity
- friction electricity
- hydroelectricity

- Piezoelectric effect is shown by certain ________ symmetric crystals. ( )
- centro
- non-centro
- mirror centro
- none

- Piezoelectric effect was discovered by _____ in 1880. ( )
- Weiss
- Thomson
- Curie brothers
- Alison

- Crystals with centre of inversion _________ exhibit piezoelectric effect. ( )
- do
- do not
- both a & b
- none

- Piezoelectric effect is a ______ effect. ( )
- reversible
- irreversible
- both
- none

- Quartz crystal_______ piezoelectric effect along the optic axis ( )
- will show
- will not show
- both a & b
- none

- The natural frequency of quartz crystal ______ with temperature. ( )
- do vary
- do not vary
- both a & b
- none

- Spontaneous polarization means, polarization of a material _______ external electric field. ( )
- without applying
- with applying
- both
- none

- The dielectric constant of a ferroelectric material increases enormously as the temperature of the material reduces to its ________ temperature ( )
- Debye
- de Broglie
- Curie
- Neel

- Ferroelectrics show ______ under the action of alternating voltages. ( )
- polarization
- polarizability
- hysteresis
- none

- The crystal structure of BaTiO
_{3}above its curie temperature is ________. ( )- cubic
- hexagonal
- rhombohedral
- triclinic

- As the temperature of BaTiO
_{3}is reduced to below curie temperature, the titanium and barium ions move up by 2.8% and oxygen ions move down by________% ( )- 5
- 3
- 2
- 1

- Ferroelectric materials are used in the manufacture of small-sized, _______ capacitance capacitors. ( )
- small
- large
- medium
- none

- The fall in permittivity of dielectric material with increasing frequency of applied electric field is usually referred to as: ( )
- anomalous dispersion
- optical dispersion
- refraction
- none

- The imaginary part of dielectric constant represent_____of the material. ( )
- dispersion
- polarization
- the dielectric loss
- none

- The dipolar polarization ceases at ________ frequency of applied electric field. ( )
- 10
^{3}Hz - 10
^{6}Hz - 10
^{11}Hz - 10
^{14}Hz

- 10
- The electronic polarization exists up to a frequency of _______. ( )
- 10
^{15}Hz - 10
^{18}Hz - 10
^{20}Hz - 10
^{22}Hz

- 10
- The real part of dielectric constant is strongly frequency-dependent and undergoes a change in sign called
___________ dispersion. ( )
- optical
- anomalous
- both a & b
- none

- High dielectric strength and high resistivity insulating materials are required to: ( )
- withstand high voltages
- reduce leakage currents
- both a & b
- none

- An insulating material used in electric machine should have sufficient mechanical strength to withstand: ( )
- vibrations
- shock
- both a & b
- none

- The relation between polarizability and dielectric constant is given by: ( )
- Clausius-Mosotti relation
- Thomson relation
- Curie-Weiss relation
- none

#### Answers

1. c | 2. a | 3. b |

4. c | 5. a | 6. c |

7. c | 8. a | 9. d |

10. b | 11. c | 12. b |

13. d | 14. b | 15. d |

16. c | 17. d | 18. d |

19. c | 20. d | 21. d |

22. b | 23. a | 24. c |

25. c | 26. a | 27. b |

28. d | 29. d | 30. a |

31. a | 32. b | 33. c |

34. b | 35. a | 36. b |

37. b | 38. a | 39. c |

40. c | 41. a | 42. d |

43. b | 44. a | 45. c |

46. b | 47. a | 48. b |

49. c | 50. c | 51. a |

#### Review Questions

- What is piezo electricity?
(Set-1–Sept. 2007), (Set-2–May 2004), (Set-2–May 2003)

- With usual notations, show that
*p = ∈*(_{0}*∈*1)_{r}−*E*.(Set-3–Sept. 2007), (Set-1–May 2004), (Set-4–Nov. 2004), (Set-1–May 2003)

- What is dipolar relaxation? Discuss the frequency dependence of orientational polarization.
(Set-3–Sept. 2007), (Set-1–May 2004), (Set-4–Nov. 2004), (Set-1–May 2003)

- Explain electronic polarization in atoms and obtain an expression for electronic polarizability in terms of the radius of the atom.
(Set-4–May 2007), (Set-4–May 2004), (Set-3–Nov. 2004), (Set-4–May 2003)

- Explain Clausius–Mosotti relation in dielectrics subjected to static fields.
(Set-1–Sept. 2008), (Set-1–June 2005), (Set-3–June 2005), (Set-3–May 2003)

- What is orientational polarization? Derive an expression for the mean dipole moment when a polar material is subjected to an external field.
(Set-1–June 2005), (Set-3–May 2004), (Set-3–May 2003)

- Obtain an expression for the internal field seen by an atom in an infinite array of atoms subjected to an external field.
(Set-1–Sept. 2007), (Set-2–May 2004), (Set-2–May 2003)

- What are the important characteristics of ferroelectric materials?
(Set-4–June 2005), (Set-2–Nov. 2004), (Set-4–Nov. 2004)

- Describe the possible mechanisms of polarization in a dielectric material.
(Set-4–June 2005), (Set-4–Nov. 2004)

- Explain the polarization mechanism in dielectric materials.
(Set-1–May 2007), (Set-1–Nov. 2003)

- What are the important requirements of good insulating materials?
(Set-1–May 2007), (Set-3–Sept. 2006), (Set-1–Nov. 2003), (Set-3–Nov. 2003)

- Explain the concept of internal field in solids and hence obtain an expression for the static dielectric constant in elemental solid dielectric.
(Set-3, Set-4–May 2006), (Set-2–Nov. 2003)

- Discuss in detail the origin of ferroelectricity in barium titanate.
(Set-3–Sept. 2006), (Set-3–Nov. 2003)

- Explain the characteristics and function of transformer oil in transformers.
(Set-2–Nov. 2004)

- Explain briefly the classification of ferroelectric materials.
(Set-2–June 2005)

- What is meant by a local field in a solid dielectric? Derive an expression for the local field for structures possessing cubic symmetry.
(Set-2–June 2005)

- Give a schematic sketch of the variation of the total polarizability of a dielectric as a function of the frequency, explaining the physical origin of the various contributions and the relevant frequency ranges.
(Set-2–Nov. 2004)

- Discuss the variation of spontaneous polarization of roschelle salt with temperature.
(Set-3–June 2005), (Set-1–May 2004)

- Obtain an expression for the static dielectric constant of a monoatomic gas.
(Set-3–June 2005), (Set-1–May 2004)

- Explain the phenomenon of anomalous dielectric dispersion.
(Set-1–May 2004)

- What is intrinsic breakdown in dielectric materials?
(Set-4–May 2007), (Set-4–May 2004), (Set-4–May 2003), (Set-3–Nov 2004)

- Explain the electrochemical breakdown in dielectric materials.
(Set-3, Set-4–May 2006)

- Obtain a relation between electronic polarization and electric susceptibility of the dielectric medium.
(Set-2–May 2007)

- What is dielectric breakdown? Explain briefly the various factors contributing to breakdown in dielectrics.
(Set-2–May 2007)

- What is orientational polarization? Explain. Obtain an expression for the mean dipole moment when a polar material is subjected to an external electric field.
(Set-2–Sept. 2007)

- Describe the phenomenon of electronic polarization and obtain an expression for electronic polarization.
(Set-3–May 2008)

- Write notes on (i) Ferro electricity and (ii) Piezo electricity.
(Set-1, Set-2, Set-3–May 2008)

- Explain the following (i) Dielectric constant, (ii) Electric susceptibility, (iii) Electric polarization and (iv) Polarizability.
- Explain the following: (i) Polarization vector and electric displacement.
(Set-4-Sept. 2008)

- Deduce an expression for Lorentz field relating to a dielectric material.
(Set-4-Sept. 2008)

- Explain the following: (i) Electric polarization and (ii) polarizability.
(Set-1-Sept. 2008)

- Describe the frequency dependence of dielectric constant.
- Derive Clausius-Mosotti equation.
- Explain the origin of diff erent kinds of polarization.
- Explain the ionic and orientation polarization.
- Explain qualitatively frequency dependence of dielectric constant.
- Explain the important requirements of insulators.
- Explain the phenomenon of ferroelectricity with particular reference to barium titanate.
- What is the frequency dependence of dielectric constant for a dielectric material?
- Explain clearly the phenomenon of ferroelectricity.
- Explain the theory of ferroelectricity and piezoelectricity.
- State and explain the terms in Clausius–Mosotti relation.
- Describe different types of polarization mechanisms.
- What are important requirements of good insulating materials?
- Write a note on piezoelectrics.
- Write in detail various types of polarization in dielectrics and derive an expression for the orientational polarization at a specified temperature.
- Derive an expression for the internal electric field in dielectrics exposed to a external electric field
*E.* - Derive Clausius–Mosotti equation of dielectrics and explain the concept of complex dielectric constant.
- Explain local field. Derive the expression for internal field for solids.
- Arrive at the relation between the dielectric constant and atomic polarizability.
- Obtain Clausius–Mosotti equation and explain how it can be used to determine the dipole moment of a polar molecule.
- How does the total polarizability depend on frequency?
- Explain piezoelectricity.
- Derive the expression for dipolar polarizability.
- Explain electronic polarization. Derive the expression for electronic polarizability.
- Explain ferroelectricity. Mention its applications.