1Dielectric and Magnetic Properties of Solids 1 DIELECTRIC AND MAGNETIC PROPERTIES OF SOLIDS Dielectric Properties of Solids Dielectrics are electrically non-conducting materials such as glass, porcelain etc, which exhibit remarkable behaviour because of the ability of the electric field to polarize the material creating electric dipoles. Dielectric Constant Faraday discovered that the capacitance of the condenser increases when the region between the plates is filled with dielectric. If C0 is the capacitance of the capacitor without dielectric and C is the capacitance of the capacitor with dielectric then the ratio C / C0 gives εr called relative permittivity or Dielectric constant. Also for a given isotropic material the electric flux density is related to the applied field strength by the equation D= ε E, Where ε is Absolute permittivity. In SI system of units the relative permittivity is given by the ratio of absolute permittivity to permittivity of free space. ε = ε0 εr ε0 is permittivity of free space. εr is relative permittivity or dielectric constant. For an isotropic material, under static field conditions, the relative permittivity is called static dielectric constant. It depends on the structure of the atom of which the material is composed. dielectrics - Polarization of dielectrics: “The displacement of charged particles in atoms or molecules of dielectric material so that net dipole moment is developed in the material along the applied field direction is called polarization of dielectric.” Polarization is measured as dipole moment per unit volume and is a vector quantity. r r P=N µ r Where µ is average dipole moment per molecule and N is number of molecules per unit volume. r r Also µ =α E where α is a constant of proportionality called polarizability. S B M Jain College of Engineering K.S. Mahesh Lohith Centre for Emerging Technologies Asst. Prof S. materials As the polarization measures the additional flux density arising from the presence of the material as compared to free space it has the same unit as D and is related to it as D =ε 0 E + P Since D = ε 0ε r E ε 0ε r E = ε 0 E + P ∴P = ε 0 (ε r − 1) E Also p = (ε r − 1) = χ ε0 E Where χ is Electric susceptibility Electrical Polarization mechanisms The electrical polarization takes place through four different mechanisms. This is the polarization of non- non-polar dielectric materials. This leads to the development of net dipole moment in the material and is the vector sum of dipole moments of individual dipoles. When the external electric field is applied all dipoles tend to align in the field direction and hence net dipole moment develops cross dielectric material. 2.This occurs through out the dielectric material and is due to the 1) Electronic polarization: separation of effective centers of positive charges from the effective center of negative charges in atoms or molecules of dielectric material due to applied electric field. Space charge polarization. Electronic polarization. Orientation polarization.2 Dielectric and Magnetic Properties of Solids 1 In Polar dielectric materials. This is the polarization of polar dielectric materials. Mahesh Lohith Centre for Emerging Technologies Asst. Prof . S B M Jain College of Engineering K. Hence dipoles are induced within the material. P = Pe + Pi + Po + Ps polarization . The net polarization of the material is due to the contribution of all four polarization mechanisms. 3. Ionic polarization. They are 1. materials In non polar dielectric materials dipoles are induced due to the applied electric field which results in the net dipole moment in the dielectric material in the direction of the applied field. 4. This occurs in ionic solids such as sodium chloride etc. E > 0. 2) Ionic polarization: Ionic solids possess net dipole moment even in the absence of external electric field. In polar dielectrics the dipoles are randomly oriented due thermal agitation. + . hence pi = N µi where N is no. + - + . + + . Prof . + . Hence the net dipole moment of the material also increases. But when the external electric field is applied the separation between the ions further increases. + + . Mahesh Lohith Centre for Emerging Technologies Asst.This occurs in polar dielectric material. + E = 0. Pe = N µ e = N α e E where is N number of atoms per m 3 ε 0 (ε r − 1) The electronic polarizability is given by α e = N polarization .S. S B M Jain College of Engineering K. + . It is found that the ionic dipole moment also proportional to the applied field strength. Polarization . of atoms per unit volume. Ionic Polarization is given by ionic dipole moment per unit volume. hence µi = α i E where α i is ionic polarisability. .3 Dielectric and Magnetic Properties of Solids 1 E<0 E>0 ⇒ electronic dipole moment = µ e = 4πε 0 R 3 E ⇒ µe ∝ E ⇒ µe = α e E where α e electronic polarizability. which possesses permanent 3) Orientation Polarization: electric dipoles. Higher the temperature more the randomness in dipole orientation smaller will be the dipole moment. Each dipole sets electric field in the vicinity. These are called localized charges. Hence the net electric field at any point within the dielectric material is given by “The sum of external field and the field due to all dipoles surrounding that point”. There fore dipole moment develops across the material. This is referred to as orientation polarization (P Po). It is denoted by Ps. E > 0.S. Mahesh Lohith Centre for Emerging Technologies Asst. This leads to the development of net dipole moment across the material. E = 0.This occurs in materials in which only a few charge carriers are capable of moving through small distances. field Expression for Internal field in case of Solids and Liquids (One dimensional ): Consider a dipole with charges ‘+q ‘and ‘–q’ separated by a small distance ‘dx’ as shown in fig. Since this is very small it can be neglected. T is absolute temperature and µ is permanent 3k T dipole moment. But when the external electric field is applied all dipoles tend to align in the field direction. Consider a point ‘P’ at a distance ‘r’ from the center of dipole. Prof . The dipole moment is given by µ = q dx. Orientation polarization depends on temperature. When the external electric field is applied these charge carriers move. During their motion they get trapped or pile up against lattice defects. S B M Jain College of Engineering K. it is Internal Field or Local Field:- polarized creating electric dipoles. point” This net field is called internal field or Local field.4 Dielectric and Magnetic Properties of Solids 1 Therefore net dipole moment of the material is zero. 4) Space charge polarization: polarization . These localized charges induce their image charge on the surface of the dielectric material. The orientation polarizability is given by µ2 α0 = Where ‘k k’ is Boltzman constant. Field When dielectric material is placed in the external electric field. The field at ‘X’ due to dipole ‘A’ is given by EXA = Er + Eθ S B M Jain College of Engineering K. E C A X B D F a a a a a a 2a 2a 3a 3a Let us find the local field at ‘X’ due all dipoles in the Array. It is given by r 2 µ Cosθ Er = 4 π ε0 r 3 2) The Tangential component or Transverse component perpendicular to the Radial r component is given by E = µ Sinθ θ 4 π ε0 r 3 Where ‘θ’ is the angle between the dipole and the line joining the dipole with the point ’P’.5 Dielectric and Magnetic Properties of Solids 1 E Er r Eθ P r θ -q +q dx The electric field ‘E’ at ‘P’ can be resolved into two components. Mahesh Lohith Centre for Emerging Technologies Asst. Consider a dielectric material placed in external electric field of strength ‘E’. ‘ε0’ is permitivity of free space and ‘r’ is the distance between the point and dipole. Prof . Consider an array of equidistant dipoles within the dielectric material. 1) The Radial Component along the line joining the dipole and the point. which are aligned in the field direction as shown in the figure.S. S. Prof . 4 π ε 0 (− a) µ ⇒ EXB = 2 π ε 0 a3 Hence the Total field at ‘X’ due to equidistant dipoles ‘A’ and ‘B’ is given by E1 = EXA + EXB µ µ E1 = 3 + 2 π ε0 a 2 π ε 0 a3 µ ⇒ E1 = π ε 0 a3 Similarly. Eθ = 0. 4 π ε 0 a3 µ ⇒ EXA = 2 π ε 0 a3 The field at ‘X’ due to dipole ‘B’ is given by E = E + E XB r θ Here r = −a and θ = 180o 2 µ Cos180o Hence Er = 3 .2 1 . the total field at ‘X’ due to equidistant dipoles ‘C’ and ‘E’ is given by E2 = E XC + E XD µ µ E2 = + 2 π ε 0 (2a)3 2 π ε 0 (2a)3 µ ⇒ E2 = (Q r = 2a) π ε 0 (2a)3 Similarly.2 µ ∴ E| = π ε 0 a3 S B M Jain College of Engineering K. E θ = 0. the total field at ‘X’ due to equidistant dipoles ‘D’ and ‘F’ is given by E3 = EXE + EXF µ µ E3 = 3 + 2 π ε0 (3a) 2 π ε0 (3a)3 µ ⇒ E3 = (Q r = 3a) π ε0 (3a)3 The net field at ‘X’ due to all dipoles in the array is given by E| = E1 + E2 + E3 + E4 + L L L µ µ µ E| = 3 + 3 + + LLL π ε0 a π ε 0 (2a) π ε 0 (3a)3 µ 1 1 = 1 + 23 + 33 + L L L π ε 0 a3 µ ∞ 1 = ∑ π ε 0 a3 n =1 n3 ∞ 1 but ∑n n=1 3 ≅ 1.6 Dielectric and Magnetic Properties of Solids 1 Here r = a and θ = 0 o 2 µ Cos0 o Hence Er = . Mahesh Lohith Centre for Emerging Technologies Asst. 2 αE ∴ Ei = E + π ε 0 a3 For Three-Dimension the above equation can be generalized by replacing 1/a3 by ‘N’ (where ‘N’ the number of atoms per unit volume).. S B M Jain College of Engineering K. Ρ and ε0 are positive quantities Ei > E. for such materials. and E is the Electric field strength.. It is given by P EL = E + 3ε 0 Clausius Clausius- usius-Mosot Mosott otti Relation: Relation: Consider an Elemental solid dielectric material. and 1.. Mahesh Lohith Centre for Emerging Technologies Asst.7 Dielectric and Magnetic Properties of Solids 1 The Local Field at ' X' is given by 1. γNαE ∴ Ei = E + ε0 γ P but polarization P = NαE ∴ Ei = E + ε0 Since γ.S..2/π by γ called Internal Field Constant...2 µ E i = E + E| = E + π ε 0 a3 Wkt µ = αE. Since they don’t posses permanent. Prof . 1.. For a Cubic Lattice γ = 1/3 and the Local field is called Lorentz field......eqn(1) Nα e 1 − 3ε 0 Where ‘N’ is the no. of dipoles per unit volume. the ionic and orientation polarizabilities are zero. αe is electronic polarizability ε0 is permittivity of free space. dipoles.. Hence the polarization P is given by P = Nα e E L Where E L is Lorentz Field P = N α e E + 3ε 0 P P = Nα e E + Nα e 3ε 0 Nα e P 1 − = Nα e E 3ε 0 Nα e E ∴P = .. ... Mahesh Lohith Centre for Emerging Technologies Asst. Ultrasonic transducers. Ferro and Piezo – Electricity : Ferroelectricity: Ferroelectricity Some dielectric materials spontaneously acquire an electric dipole moment below a certain temperature. Similar to Ferromagnetic materials Ferroelectric materials also exhibit ferroelectric hysteresis... Ferroelectric hysteresis is the lagging of the polarization with respect to applied electric field Strength during the positive polarization and negative polarization of the specimen.....S. eqn ( 2) Where ‘D’ is Electric Flux Density and ε r is Dielectric Constant. Prof . microphones.... Lithium Niobate and Barium Titanate are the few examples of ferroelectric materials. Using the above relation if the value of dielectric constant of the material is known then the electronic polarizability can be determined. Equating equations (1) and (2) Nα e E ∴P = = ε 0 (ε r − 1) E Nα e 1 − 3ε 0 Nα e E Nα e = 1− ε 0 (ε r − 1) E 3ε 0 Nα e E Nα e + =1 ε 0 (ε r − 1) E 3ε 0 Nα e 3 + 1 = 1 3ε 0 (ε r − 1) Nα e ε r − 1 ∴ = 3ε 0 εr + 2 The above equation is called Clausius – Mosotti relation.8 Dielectric and Magnetic Properties of Solids 1 The polarization is related to the applied field strength as given below D =ε0 E + P Since D = ε 0ε r E ε 0ε r E = ε 0 E + P ∴P = ε 0 (ε r − 1) E .. Quartz.... S B M Jain College of Engineering K. This Is referred to as spontaneous polarization. Analogy with magnetic material results in a type of dielectric materials called ferroelectric materials. Ferroelectric materials exhibit Piezo- electricity and Pyro-electricity. T T −θ is the temperature and θ is a temperature very close to a temperature E called Curie temperature (Tc).. It is a plot of polarization (P) versus Electric field strength (E).. The static dielectric constant of a ferroelectric material changes P with temperature which is given by ε r = C Where C is a constant. Infrared detectors and capacitors. Ferroelectric Ferroelectric Hysteresis materials are used in pressure transducers. The inverse of Piezoelectricity is called electrostriction. In the crystal the distribution of the ionic charges about their lattice sites is symmetrical. then the polarization can follow the alternations of the field and hence the dielectric constant remains static. The piezoelectric crystals are used in electro-mechanical transducers. Dielectric loss & complex dielectric constant): constant):- It is found under alternating field conditions of high frequency. the dielectric constant is a complex quantity. When crystals like Tourmaline.S. When dielectric materials are placed in alternating field the polarization tend to reverse as the polarity changes. If the frequency of the field is low (less than 1M Hz). The charges produced on the surface due to stressing are proportional to the applied force which is utilized in the conversion of mechanical energy into electrical energy.9 Dielectric and Magnetic Properties of Solids 1 Piezoelectric Materials: Materials Certain dielectric materials are electrically polarized when their surfaces are stressed. But when the crystal is stressed the symmetry is altered due to the displacement of charges which results in non zero internal field. Hence the piezoelectric materials are also called electro-strictive materials. Piezoelectric strains are very small. Mahesh Lohith Centre for Emerging Technologies Asst. Rochelle salt and Quartz are sliced in a particular fashion they exhibit piezoelectric effect. as Oscillators to generate highly stable frequency and measurement of velocity of ultrasonics in solids and liquids. Prof . Space Charge Polarization Orientation Ionic Electronic Frequency S B M Jain College of Engineering K. This phenomenon is called piezoelectric effect and the materials are called piezoelectric materials. Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric constant. For quartz for a strain of the order 10-7 the corresponding electric field is 1000v/cm. Thus the net internal field is zero. Electrostriction is a phenomenon of straining a crystal by applying an electric field. Hence the corresponding electric fields are very high. 10 Dielectric and Magnetic Properties of Solids 1 Under alternating field conditions of high frequency (Greater than 1MHz) the oscillations of dipoles lag behind those of field.εr’’. 6) Piezoelectric crystals are used in oscillators and vibrators. Complex Dielectric Constant: Constant .In the alternating field conditions during the rotation of dipoles they have to overcome some sort of internal friction. Hence as the frequency of the field increases the polarization decreases and hence the dielectric constant decreases. which is dissipated as heat by the material. 3) Mica and Asbestos insulation is provided in electric Iron in order to prevent the flow of electric current to outer body. 4) Varnished cotton is used insulators in transformers. The ionic polarization ceases in the frequency range 1011 Hz to 1014Hz. εr’ determines the component of current out of phase by 90° with the field.S. Important applications of Dielectric Materials: Dielectric materials find a wide range of applications as insulating materials. Finally only electronic polarization remains. This occurs in the frequency range 106 Hz to 1011Hz. Where εr’’ determines Dielectric Loss. If the frequency is increased further they are completely unable to follow the alternations in the field and hence the molecular process Orientation polarization ceases due to dielectric relaxation. S B M Jain College of Engineering K. Mahesh Lohith Centre for Emerging Technologies Asst. Prof .The complex dielectric constant is given by εr* =εr’ . This is known as Anomalous Dielectric Dispersion.As the frequency is increased further other polarizing mechanisms start to cease one after another. 1) Plastic and Rubber dielectric are used for the insulations of electrical conductors 2) Ceramic beads are used for the prevention of electric short circuiting and also for the purpose of insulation. Dispersion Dielectric Loss: Loss . 5) Dielectric materials are used in the energy storage capacitors. This is called as dielectric loss. Langevin was able to derive an expression for the diamagnetic susceptibility and is given by − µo n e2 ∑r i i 2 χ= 6m Here χ is diamagnetic Susceptibility µo is permeability of free space. Hence atom acquires an effective magnetic moment which is opposite to the applied magnetic field. When diamagnetic materials are placed in the magnetic field. The magnetic dipole moment of the atoms of the material is zero. while the other accelerated. But the field is able to modify the motions of the electrons in orbits which are equivalent to tiny current loops. explained diamagnetism from the electron theory of matter. A perfect diamagnet expels all the flux lines from the interior of the material. does not tend to align the atoms (dipoles) of the substance. The atoms in diamagnetic materials contain even number of electrons. Bismuth. Prof . in 1905. 1) Diamagnetic materials 2) Paramagnetic materials 3) Ferromagnetic materials 4) Antiferromagnetic materials 5) Ferrimagnetic materials Diamagnetic Materials: Materials Diamagnetic materials are the substances which become weakly magnetized in the direction opposite to that of the applied field.S. Mahesh Lohith Centre for Emerging Technologies Asst. The electrons in each pair have orbital motion as well as spin motions in opposite sense. e is electronic charge. m is the mass of the electron and ri is the radius of the ith orbit. Gold. The electron moving in a direction so as to produce a magnetic field in the same direction as the external field is slowed down. Water are the few examples for diamagnetic materials. When diamagnetic materials are placed in the magnetic field they expel the flux lines. Hence it is universal property of matter. S B M Jain College of Engineering K. Langevin’ Langevin’s theory of Diamagnetic Materials: Materials Langevin. It is not affected by thermal agitations and hence independent of temperature. They do not posse’s permanent magnetic dipoles. The resultant magnetic moment of the atom is thus zero. The electrons in such atoms are paired. The Lenz’s law acts in the atomic scale. Diamagnetism is present in all magnetic materials and is outweighed in case of the Diamagnetic materials.11 Dielectric and Magnetic Properties of Solids 1 Magnetic Properties of Solids Classification of magnetic materials: materials can be classified in to five types on the basis of magnetic dipoles. Hence the net dipole moment decreases with temperature. Materials composed of elements Iron. Platinum. The expression for paramagnetic susceptibility as derived by Langevin is µ0 n pm2 χm = 3k T χm is the paramagnetic susceptibility. When such materials are placed in the magnetic field they get weakly magnetized in the direction of the applied magnetic field. Nickel and Cobalt exhibit ferromagnetism. In the absence of S B M Jain College of Engineering K.12 Dielectric and Magnetic Properties of Solids 1 Paramagnetic Materials: Materials Paramagnetic materials posse’s permanent magnetic dipoles. 1) The neighboring dipoles in ferromagnetic materials interact and they group together to form ferromagnetic domains. In the absence of the external magnetic field all dipoles tend to orient in random directions due to thermal agitation and hence the net magnetic moment of the material is zero. atoms or molecules of a paramagnetic material have a net intrinsic paramagnet magnetic moment due to the spin and orbital motions of electrons in it. In ferromagnetic materials the interaction between the neighboring dipoles is very strong. The Weiss theory is based on the following assumptions. Langevin’s theory theory of Paramagnetism: Paramagnetism Langevin explained paramagnetism on the basis of. Ferromagnetic Materials: Ferromagnetic materials can be strongly magnetized by relatively weaker magnetic fields. Aluminium. Manganese are the few examples of paramagnetic materials.S. the dipoles experience torque and they tend to align in the applied field direction. This interaction is called spin exchange interaction. Due to thermal agitation of the molecules the average alignment produced give rise to the net magnetic moment of the material in the direction of the applied field. Higher the temperatures more will the randomness in the orientations of the dipoles due to thermal agitations. When paramagnetic materials are placed in the magnetic field. k is Boltzmann constant. Each domain is magnetized to saturation. In paramagnetic materials the interaction between the neighboring dipoles is extremely week. µo is the permeability of free space. Prof . The paramagnetism is a function of temperature. Mahesh Lohith Centre for Emerging Technologies Asst. Weiss molecular molecular field theory of Ferromagnetism: Ferromagnetism In the year 1907 Weiss proposed a theory for susceptibility of ferromagnetic materials. pm is magnetic moment. T is absolute temperature. For temperatures less than ferromagnetic Curie temperature the dipole-dipole interactions are very strong which overcome the thermal agitations. 2) The cause of spontaneous magnetization within each domain is the existence of the molecular field which tries to produce a parallel alignment of the dipoles.13 Dielectric and Magnetic Properties of Solids 1 external magnetic field.S. T is the temperature above ferromagnetic curie temperature (Tc). since the direction of magnetization of different domains is different. B-H curve is a plot of the induction flux density versus the applied field strength during magnetization and demagnetization of a ferromagnetic specimen. Hence the susceptibility is very high. Mahesh Lohith Centre for Emerging Technologies Asst. the average resultant magnetization of ferromagnetic material is zero. Ferromagnetic Hysteresis (B- (B-H Curve): Curve) During the magnetization and demagnetization of the specimen the induction flux density (B) lags behind the applied magnetic field strength (H) and is called ferromagnetic hysteresis. Prof . B Bs a b c o f H e d S B M Jain College of Engineering K. Based on the above assumptions Weiss was able to derive an expression for ferromagnetic susceptibility as a function of excess of temperature which is given by Nµ 2 C χ= = 3k (T − θ ) T − θ Here χ is Ferromagnetic Susceptibility. But for temperature higher than ferromagnetic Curie temperature the violent thermal agitations overcome dipole interactions and the susceptibility varies inversely as excess of temperature and the material is transformed into paramagnetic phase. C is Curie constant. During the magnetization of the ferromagnetic specimen by applying external magnetic field the magnetization of the specimen takes place either due to domain wall movement under weak fields or to the rotation of domain’s magnetic moments under applied strong magnetic fields. θ is a temperature very close to Tc. Ceramic magnetic materials (ferrites): . They posses low coercivity S B M Jain College of Engineering K. Further the complete variation of H from +H to –H and back from –H to +H will trace a closed hysteresis curve abcdef. The magnet. If H is reduced the domains tend to reorient to their original direction and B decreases. B Properties: Properties - 1.14 Dielectric and Magnetic Properties of Solids 1 The origin O represents the initially unmagnetised specimen and zero magnetic intensity. But when H reduces to a value zero. They can be easily magnetized and demagnetized.Which are made of metals 2. Based on their properties each type is classified into Hard and Soft magnetic materials. The corresponding field Hc is called Coercive field.. As H is increased B also increases and saturation (Bs) is reached (curve oa). Metallic magnetic materials: materials . For an applied reversed field of magnitude Hc (oc) the material is completely demagnetized and B=0(Curve bc). E. Classification of magnetic materials into Hard and soft: soft Magnetic material are further classified into two types 1. When the magnetizing force is removed the magnetic flux retained by the specimen is measured as retentivity. steel. They posses high susceptibility 3. Soft iron. They posses high permeability H 2. 1) Soft metallic magnetic materials: -They is also called “permeable permeable magnetic materials” materials since they allow the flux lines to pass through very easily.g. ivity If further H is increased the curve cd is traced and the reverse saturation is achieved.S. Metallic Magnetic Materials: Materials: They are classified into two types. Mahesh Lohith Centre for Emerging Technologies Asst. is called soft magnet. which is made of soft magnetic material. At saturation all domains are oriented in the same direction as the applied field. This is called as Hysteresis Loss. When H is varied from its negative maximum value. retentivity Now H is reversed in direction and increased. A magnet material that retains magnetic field is called magnet.Which are the compounds of metals and non- metals. The area enclosed by the curve abcdef gives the energy loss per cycle per unit volume of the material. The flux density corresponding to H=0 is called remnant induction or remnant flux density (Br=ob). to its original positive maximum value then a similar curve defa is traced. B does not reduce to Zero and curve ab is traced. through zero. Prof . The measure of magnetic intensity required to completely demagnetize the material is called Coercivity Coercivity. Mahesh Lohith Centre for Emerging Technologies Asst. They are used in magnetic tapes. Since the area enclosed by the Hysteresis curve is large. They posses high saturation magnetization 6. They are called “hard hard magnetic materials” materials because they cannot be magnetized and demagnetized very easily.g. the energy loss per cycle is also high. They are used in A C current machinery 3. They are used in audio and video transformers materials: . They posses low retentivity 5.They are also called as “permanent 2) Hard magnetic materials: permanent magnetic materials” materials because of their ability to retain magnetic field even after the magnetizing force is withdrawn. High magnetization force is needed for saturation. They are used for making permanent magnets 3. E. They are used in electromagnets 2. They are used in digital computers H 2. Properties: Properties . Alnico. They posses high retentivity 5. Applications: Applications - 1. Nickel steel. They posses high coercivity 4. They are used in Communication equipments 4. When a Hard magnetic material is magnetized it is called Hard magnet. They are used in transducers 4. 6.(compared to soft magnetic materials) 1. the energy loss per cycle is small.15 Dielectric and Magnetic Properties of Solids 1 4.S. They posses low permeability 2. Since the area under the Hysteresis curve is very small. B Applications: Applications - 1. Prof . S B M Jain College of Engineering K. They posses low susceptibility 3.