Electrical Engineering Materials lecture notes

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ELECTRICAL ENGINEERING MATERIAL LECTURE NOTES for Bachelor of Technology in Electrical Engineering & Electrical and Electronics Engineering Department of EE & EEE Veer Surendra Sai University of Technology (Formerly UCE, Burla) Burla, Sambalpur, Odisha Lecture Note Prepared by: Prof. Ramesh Chandra Prusty SYLLABUS ELECTRICAL ENGINEERING MATERIALS (3-1-0) Credit-04 MODULE-I (10 HOURS) Conductivity of Metal: Introduction, factors affecting the resistivity of electrical materials, motion of an electron in an electric field, Equation of motion of an electron, current carried by electrons, mobility, energy levels of a molecule, emission of electrons from metals, thermionic emission, photo electric emission, field emission, effect of temperature on electrical conductivity of metals, electrical conducting materials, thermal properties, thermal conductivity of metals, thermoelectric effects. MODULE-II (10 HOURS) Dielectric Properties: Introduction, effect of a dielectric on the behavior of a capacitor, polarization, the dielectric constant of monatomic gases, frequency dependence of permittivity, dielectric losses, significance of the loss tangent, dipolar relaxation, frequency and temperature dependence of the dielectric constant, dielectric properties of polymeric system, ionic conductivity in insulators, insulating materials, ferroelectricity, piezoelectricity. MODULE-III (10 HOURS) Magnetic properties of Materials: Introduction, Classification of magnetic materials, diamagnetism, paramagnetism, ferromagnetism, magnetization curve, the hysteresis loop, factors affecting permeability and hysteresis loss, common magnetic materials, magnetic resonance. MODULE-IV (10 HOURS) Semiconductors: energy band in solids, conductors, semiconductors and insulators, types of semiconductors, Intrinsic semiconductors, impurity type semiconductor, diffusion, the Einstein relation, hall effect, thermal conductivity of semiconductors, electrical conductivity of doped materials. BOOKS 1 C.S.Indulkar and S. Thiruvengadam, S., “An Introduction to Electrical Engineerin 2 Kenneth G. Budinski,, “Engineering Materials: Prentice Hall of India, New Delhi MODULE-I CONDUCTIVITY OF METALS INTRODUCTION: The most important properties of metals are their high thermal and electrical conductivities. Silver has the highest electrical conductivity. Copper comes next and is similar to silver from the point of view of atomic structure ; both belonging to the same group of periodic table. The conductivity of copper is less than that of silver. Since supplies of copper are not abundant in nature, aluminium which is light and has a high conductivity is rapidly becoming more important as a conductor material. Gold which has a conductivity higher than that of aluminium but lower than that of silver or copper does not find use in electrical industry because it is expensive. Metals having complex structures such as As, Sb, Bi, Sn, Hg have lower conductivities which lie between those of ideal metal (very high conductivity) and of insulators (negligible conductivities). FACTORS AFFECTING THE RESISTIVITY OF ELECTRICAL MATERIALS 1. Temperature : The electrical resistance of most metals increases with increase of temperature while those of semiconductors and electrolytes decreases with increase of temperature. Many metals have vanishing resistivity at absolute zero of temperature which is known as superconductivity. 2. Alloying : A solid solution has a less regular structure than a pure metal. Consequently, the electrical conductivity of a solid solution alloy drops off rapidly with increased alloy content. The addition of small amount of impurities leads to considerable increase in resistivity. 3. Cold Work : Mechanical distortion of the crystal structure decrease the conductivity of a metal because the localized strains interfere with electron movement. 4. Age Hardening : It increases the resistivity of an alloy. MOTION OF AN ELECTRON IN AN ELECTRIC FIELD In a conductor, the electrons are moving about with random velocity , the magnitude of which depends upon the temperature. There are two comonents of motion, as follows : 1. Random motion , due to thermal effects. 2. Directed motion , the direction being determined by the polarity of the electric field. EQUATION OF MOTION OF AN ELECTRON When no electric force is applied , the free electrons move about through the conductor in a random manner in such a way that the number of electrons moving from right to left is the same as the number moving from left to right and the resultant current is nil. If an electric force is applied to the conductor, each electron has superposed on to its random motion, a motion impressed on it by electric force, and the electrons as a whole are driven through the conductor by the continued action of this electric force. THE CURRENT CARRIED BY ELECTRONS In a current carrying conductor, the electrons drift along with an average velocity which is generally small compared with their random velocity due to thermal agitation. Let a current I be carried along a conductor of cross section A by electrons of charge -e and of average drift velocity v. In time dt the electrons will travel a distance vdt and the number of electrons crossing any cross section A in time dt will be the number contained in the volume Avdt. Thus if there are N electrons per unit volume of the conductor the total charge flowing through the section in time dt is dq= -e.N.A.v.dt I=dq/dt= -eNAv 2 And current density I/A= -e Nv = +e NEt/m Since, v= -eEt/m The expression for current density shows that the current density does not depend on the size of the conductor. It is a general property of the material. Finally, the current density is proportional 2 to the electric field strength and the constant of proportionality e Nt/m is called conductivity of 2 the material and is denoted by σ= e Nt/m . Ohm’s law follows as an immediate consequence of the relation J= c ; because I=J.A = σE.A = (σV/l).A Where l is the length of conductor and V is the voltage applied to the conductor ends. Since σA/l=R where R is the resistance of the conductor. MOBILITY : It has been noted that the average drift velocity of the electrons in an applied field is proportional to the field , the absolute magnitude of the proportionality factor et/m, being called the mobility of the electrons which is denoted by u. The mobility may thus be defined as the magnitude of average drift velocity per unit field. The mobility and the conductivity are related by the equation σ= NeU. Thus the mobility of the electrons can be determined by knowing the conductivity of the material and estimating the number of free electrons. Mobility, U= σ/Ne = 1/ρNe -8 28 -19 =1/(1.73x10 x8.5x10 x1.6x10 -3 2 =4.25x10 m /volts-sec . The order of magnitude of collision time for copper atoms may be determined from the relation 2 t=m/Ne ρ . In the absence of an electric field no electric current is observed in the conductor. When an electric field is applied to the conductor the electrons moving in the direction of the electric force acting on them retarded. Thus the movement of electrons in the direction of the field force predominates over that which proceeds in the opposite direction, the result being an electric current. Taking account of fact that only quantum states of motion are possible for electrons, the acceleration may be conceived as the transfer of an electron into a new quantum state of greater velocity and the deceleration as the transfer of an electron into a state of less velocity. The electric current may thus be treated as the predominance of states that corresponds to the motion of electrons from one end to another over the opposite states. In actuality , the movement of electrons among the atoms of a solid is far more complex than in vacuum and could be taken account . In a perfectly built crystal the electrons can move in nearly the same way as in vacuum. But crystal irregularity due to impurities and thermal agitation distorts the regular flow of charges and produces a electric field that force the electrons out of their initial path. The electrons move freely only in undistorted section of the crystal. Here they accumulate energy with the increase in speed. ENERGY LEVELS OF THE MOLECULE: The energy of an atom changes only by a single means, i.e. , a change occurs when one electron passes from one quantum states to another. However, the energy of a molecule is likely to change in either one or all of the following three ways: 1. The energy changes of a molecule may take place like that of an atoms. 2. Since the atoms of a molecule vibrate with respect to one another, the vibrational energy of molecule may also assume discrete values. 3. Furthermore since the molecule rotates as a whole , the rotational energy is also quantisedand a change in the state of a molecule may result in a change in rotational energy. The energy states of a molecule are therefore described by indicating the state of its electronic cloud (electron level) the state of its vibrational motion (vibrational level) and the state of its rotational motion (rotational level). . The difference between the rotational levels are smaller than those between the vibrational levels. Further differences between the vibrational levels are smaller than those between the electronic levels. His corresponds to a type of house numbering system. Suppose the electronic levels in a molecule are at 100, 200, 300, … units, the vibrational levels are at 10,20,30, … units and the rotational levels are at 1,2,3, … units . In such case, a molecule in the first electronic level, the second vibrational level and the third rotational level will have a total energy of 123 units. EMISSION OF ELECTRONS FROM METALS If an electron acquires excess energy at least to the work function by some mean, it will escape from the metal and will travel to a nearly electrode held at a positive potential with respect to the emitting surface. A continuous flow of such electrons constitutes the thermionic emission of vacuum tubes. Electrons may acquire sufficient energy to escape from the metal in either of the following ways: I. By heating of the metal: Thermionic emission II. By projecting light of sufficiently small wavelength on the metal: photoelectric emission. III. By applying an intense electric field to the metal: Field emission ,and IV. By bombarding the metal surface with electrons: secondary emission. THERMIONIC EMISSION Thermionic emission is a process of evaporation which takes place when some of the conduction electrons in the cathode have enough kinetic energy to escape through its surface. The emission therefore, depends upon the distribution of energy among the free electrons in the cathode. This distribution of energy is a function of temperature. To escape from the metal, an electron must have a component of velocity at right angles to the surface, and the corresponding kinetic energy, must be at least equal to the work done in passing through the surface. This is denoted b the energy corresponding to the escape level. , Since the free electrons in a metal move at random, not all of those whose total kinetic energies exceed the will have sufficient energy in the right direction to escape. Only a certain , proportion will do so and these will constitute the thermionic emission. In the figure the number of electrons having kinetic energies equal to or greater than is the number occupying the energy levels of that value and all higher values. The number is proportional to the area under the corresponding part of the distribution curve, which has been heavily shaded in the figure. Consequently, the thermionic emission from the metal is proportional to the heavily shaded area, i.e., at a given temperature, the number of electrons represented by the shaded area will be able to leave the metal surface. If the ‘tail’ of the curve does not exceed beyond the value the thermionic emission will be zero. Once this point has been reached the emission rapidly increases with temperature. Since even at the absolute zero temperature many electrons in the metal have high energies, up to the value , the minimum excess energy necessary for thermionic emission is given by = . The quantity is called the work function of the metal. The existence of a work function implies that there are forces which restrain an electron from escaping as it approaches the surface of a metal. The emission current is strongly dependent upon the work function. The larger the values of work function, more difficult it is for the electrons to escape from the metal. In other words, if the difference between Fermi level and the escape level is large, the emission is relatively small. The value of work function can be estimated from measurements of emission current as function of temperature. The choice of cathode material in vacuum tubes is based on ease of electron emission(or lo electron affinity) so that an adequate supply of electrons is obtained with low operating temperatures. The close parallel between thermionic emission and evaporation was recognised by Richardson who showed that the current emitted per unit area of the metal should be given by () = Where is the work function of the metal, K is the Boltzmann constant and T is the absolute temperature. A is called the thermionic emission constant and should be a universal constant for all metals, equal to 120Amp/ / , but its value are found to vary considerably, an effect which is generally attributed to partial reflection of electrons at the surface of the metal. The emission obtained from different faces of a metal is also found to vary and the vary and the value of A for polycrystalline materials is quite low. The number of material available for use as cathodes in thermionic values is greatly limited by the requirement of high electron emission at temperatures where the material does not disintegrate. The temperature at which adequate emission is obtained is determined primarily by the value of the work function . The common materials used for cathodes are tungsten, thoriated tungsten and a mixture of barium oxide-strontium oxide. In other to obtain a current density of about 1A/ the operating temperatures for these materials are approximately 2500, 1900 and 1100 K respectively. Tungsten is resistant to ‘poisoning by residual gas and gives long live to transmission tubes. Thoriated tungsten is not so resistant to ‘poisoning’ but has considerable advantage because of lower operating temperature. The barium strontium oxide cathode is also liable to be ‘poisoning’ by the presence of residual gas. It is therefore necessary to maintain high vacuum in tubes where it is used. Table 3.1 gives values of the thermioninc emission constant, A ,work function and Fermi energy, for a number of metals, semi conductors oxide coated cathodes. PHOTOELECTRIC EMISSION This involves the interaction of light with electrons showing that light has quantum qualities. In photoelectric emission, a beam of light of frequency , v interacts with the electrons only in discrete quantities of energy, hv where h is planck’s constant. The electrons have to acquire additional energy in order to reach the escape level, . In other words, emission is possible only if an individual quantum of light has sufficient energy above the escape energy. The necessary condition of emission is hv≥ e where is expressed in electron volts. For copper, the work function, is very high and potassium it is very small. Hence, in the case of potassium, photoelectric emission is possible for relatively low light frequencies. Measurement of the minimum frequency(given by hv=e ) for which photoelectric emission can occur gives a method of estimating the work function. When the frequency is greater than the minimum, the maximum energy which the electrons may possess outside the metal is (hv- e ) joules. In practice, a frequency greater than the minimum is usually used, a retarding potential being applied to the collecting anode. The quantity of energy transferred to an electron in the material is determined only by the frequency of the light vibrations and is independent of the intensity of light ray. As the intensity of light is increased, the number of light absorbing electrons increases, but the energy absorbed by each of the electrons remains unchanged. Light is capable of transferring electrons to the free state inside a material thus increasing the electrical conductivity of the material of the material. When the energy imparted to the electrons is quite large, the later may be emitted from the material into the surrounding medium. This phenomenon is known as the thephotoemissive effect, or photoemissitivity, whereas the increased electrical conductivity produced by light is called the photoconductive effect, or photoconductivity. The study of photoelectric phenomena provides us with information about the properties of electrons in materials such as the amount of energy required to transfer an electron into the free state. A part from photoconductivr and photoemissitivity cells, a third type-the photovoltaic or barrier layer cell exists. In the photovoltaic cell, light establishes an emf between two substances such as a layer of cuprous oxide on copper or of selenium or iron. Photovoltaic cells are widely used in illumination and exposure meters. Photoemissive cells or photo-tubes are two-elements tubes whose cathodes emit electrons when exposed to light. The anode current resulting from a given amount of incident light is a function of the wavelength of light, which gives rise to a number of useful application. These include door openers, counters, position and temperature control and colour analysis. The materials most commonly used for photoconductive cells in the visible part of the spectrum are selenium, cadmium sulphide and thallic sulphide and those for infrared rays are lead sulphide, lead selenide and lead telluride. Such cells are widely used in automation and for remote control industrial processes. FIELD EMISSION Field emission and cold emission occurs when the direction of the applied electric field is such that it attracts electrons out of the metal. The potential energy at the surface of the metal is(+ ). This is the energy required for the extraction of the electron from the metal. When the external field is large, the energy required to extract the the electrons from the metal is small, which shows the variation of potential energy with distance. EFFECT OF TEMPERATURE ON THE ELECTRICAL CONDUCTIVITYI OF METALS As the temperature is increased there is a greater thermal motion of the atoms which decreasesthe regularity in the atom spacings with a consequent decrease in the mobility of the electrons. The resistivity of most metals therefore increase with an increase in the temperature. The electrical conductivity of a metal on the basis of the free electron model is given by c= = , where is the mean free path and equals e where c is the mean velocity of the electrons. The value of c corresponds to the Fermi energy because only those electrons which are at top of the Fermi distribution curve can be accelerated and can gain energy. This velocity is of the order of 10 cm/sec for most metals and since = 10 sec at room temperature, the mean free path is of the order of 10 cm, or about 100 times the atomic spacing in solids. Since the number and the energy of the electrons at the top of the Fermi distribution curve vary insignificantly with temperature, the change in temperature must be associated with a change in the mean free path. Ideally the mean free path of an electron in a perfectly regular lattice, each electron will exist in a particular energy state and thus will have a fixed velocity indefinitely. Practical metals do not have a perfect lattice because of impurities and because of the deviation of atoms about their mean position due to lattice oscillations (Debye waves). Thus the mean free path for an imperfect lattice is finite. This accounts for the lower conductivities of alloys which have a disordered lattice. The lattice imperfections due to impurities and atomic oscillations cause scattering of the electron waves which is analogous to the scattering of light waves in an imperfect crystal. The scattering is independent of temperature giving rise to the constant resistance which is characteristic of alloy materials. Since the lattice oscillations decrease at low temperature the scattering of electron waves falls and the conductivity therefore increases rapidly as the temperature approaches absolute zero. There is a limiting value beyond which the conductivity will not increase, the limit being determined by the previous history of the metal. In general, the purer the specimen, the higher is the limiting conductivity. The conductivity of many metals decreases linearly as the temperature is increased above the room temperature but below this temperature the conductivity increase markedly and with a higher power of the absolute temperature( ). ELECTRICAL CONDUCTING MATERIALS COPPER: Pure annealed copper is used for the winding of electrical machines. High purity copper is obtained by electrolytic refining. Traces (0.1%) of iron, silicon or phosphorous seriously reduce the conductivity of copper. The conductivity of copper is also decreased when it is hard drawn into wires for use in machines. Annealing is therefore necessary before the material can be used in machines. Hard drawn copper because of its increased mechanical strength compared with annealed copper is used for conductors in low voltage overhead distribution lines. Long span lines of thin cross section require conductors of higher mechanical strength. This is achieved by adding a small percentage of cadmium to copper. Cadmium increases the mechanical strength of copper without affecting its conductivity adversely. The usual addition of cadmium are between 0.8% and 1%. Copper conductors having a steel core are also employed for long span transmission lines, where a combination of high conductivity, small sag and minimum cross section are desired. In such conductors, an insulating tape over the wire has to be provided in order to prevent the corrosive action of steel on copper. For ordinary insulated stranded cables V.I.R insulation is almost universally employed. In such cables the conductor stands are tinned in order to protect the copper from the sulphur of the V.I.R. The tinning process assists in soldering and operations. Copper is employed in machine windings because it is easily workable without any likelihood of fracture. Further, it can be soldered easily thus simplifying the jointing operation. ALUMINIUM: Aluminium conductors are particularly suitable for operations in very high ambient temperatures. Use of aluminium as an electrical material particularly in the aircraft industry has considerable advantages because of the saving in weight involved. Again electrochemical plants are enormous user of aluminium bush bars. This is because electrolytic cells operate with heavy current wit low voltages and to carry these currents massive bars are required. Aluminium because of its lightness is being used more and more for such bush bars. The current carrying capacity of aluminium being 75% that of copper and its density being approximately one-third that of copper an aluminium bush bar is only half the weight of copper bush bar of equal current carrying capacity. Since aluminium costs a little less than copper, an aluminium bush bar will cost only about half as much as its copper counterpart. The steel reinforced aluminium conductor (A.C.S.R.) is extensively being used for long span transmission lines. In the commercial form aluminium is obtainable with a purity of about 99% but it is generally alloyed with small quantities of copper, zinc, nickel or magnesium to improve its hardness and strength. Aluminium is not easily solderable but fluxes have been devised to make soldering easy. Mechanical clamping and screwing methods have also been developed. TUNGSTEN: Tungsten has the highest melting point among metals. It is therefore suitable for applications requiring high operating conditions, such as lamp and valves filaments. The resistivity of tungsten is 5 Ω-cm which is twice as poor as that of aluminium. However the great hardness and the high boiling point and melting points of tungsten coupled with its resistance to abrasion. Establish this metal as an outstanding material for electrical contacts in certain applications. It is extremely resistant to the destructive forces of arcing. Typical operating conditions for tungsten contacts are: Voltage a.c. or d.c.upto 230V Current upto 15A Typical applications of tungsten contacts are in battery ignition systems, vibrators are electric razors. CARBON AND GRAPHITE: The severity of sparking and the rate of commuter wear in electrical machines is greatly reduced by using brushes mad of carbon. Carbon is also used in automatic voltage regulators for making the pressure sensitive pile resistors. Among other uses of carbon are for making arc wielding electrodes, fixed and variable resistors for light currents and contacts of certain classes of d.c. switchgear which are subjected to arcing. The action of carbon in a microphone is that of providing a material, the resistance of which decreases when it is compressed. The resistance temperature coefficient of carbon is negative. IRON AND STEEL: Steel is employed as conductor rail in traction on account of its cheapness and rigidity. Galvanised steel and iron wires which are generally used for earth conductor in low voltage distribution system may also be used for the phase conductors in rural areas where cheapness is the main consideration. Such lines will however have large voltage drops because of the high resistance and inductance. Addition of manganese has a hardening effect on steel and manganese steel(about 13% manganese) has the further property of being practically non magnetic . Steel alloyed with chromium and aluminium is used for making starter rheostats where lightness combined with robustness and good heat dissipation are important considerations. Cast iron is used in the manufacturing of “resistance grids” to be used in the starting of the large dc motors. NICKEL: The material is used extensively for making the electrodes of thermionic valves, and sparking plugs. It is also, used to form the positive plate of the Nife accumulator which has distinct advantages over the ordinary lead acid accumulator. LEAD: Lead has two important electrical applications. It is used to form(a) cable sheaths and (b) the plates of lead acid accumulator. Lead sheaths are required to protect the insulation of the cable from effects of moisture. TIN: The important electrical use of tin is in the manufacture of low current fuses. ALLOYS: Alloy materials are used for making resistors for laboratory instruments and for laboratory standards where a high constancy of resistance is desirable. They are also used for making heater and thermo-couple elements. The important alloys are: a. Constantan or Eureka(55-60%) Cu, (45-40%) Ni b. German silver (an alloy of Cu,Zn and Ni) c. Manganin (86% Cu, 2% Ni , 12% Mn) d. Nichrome (60% Ni, 15% Cr, 24% Fe) ELECTRICAL CONTACT MATERIALS: Several elements, in their relatively pure form such as copper, molybdenum , nickel, palladium, silver and tungsten are acceptable make and break contact materials. Alloys and heterogeneous mixtures which are, in general, combinations of the elements mentioned above are also used in electrical contacts. Silver is an important contact material. Copper added to silver reduces the cost of the contact material, whereas a combination of tungsten and silver results in a contact material having the advantages of the individual metal. A silver tungsten contact material will have high thermal and electrical conductivity, low contact resistance and high resistance to oxidation due to the presence of silver and a high melting point and high resistance to electrical erosion due to the presence of tungsten. The principal deficiencies of copper as a contact material are its poor resistance to oxidation and the relative case with which it forms other chemical compounds (e.g.,sulphides) that interface with its performance. If however, the frequency of operation is not too low, and if there is some wiping action between the contacts, copper contacts may be used at currents (a.c. or d.c.) of up to about 500A and voltages (a.c. or d.c.) of up to about 600 volts. Typical applications of copper contacts are in control relays, motor starter switches and tap changers. Contacts made of silver and silver alloys are possibly the most widely used in electrical industry. Silver is far superior to copper in its resistance to oxidation and it exhibits low contact resistance. Silver and silver alloy contacts may be used for voltage(a.c. or d.c.) upto 600 volts and direct currents upto 50A and alternating currents upto 200A. Such contacts are used in all types of industrial, relays, generator cut outs, thermal overload devices and thermostatic control. NON-LINEAR CONDUCTORS: Certain conducting materials do not obey Ohm’s law and the resistance of such materials may vary with the applied voltage. Such material are said to possess non linear resistance. There are other classes of materials in which the resistance varies not only with the applied voltage but also with the polarity of the applied voltage. Such materials are said to possess rectifying properties. THERMAL CONDUCTIVITY OF METALS : It is observed that metals which are good conductors of electricity are also good conductors of heat. When a homogenous isotropic materials is subjected to a temperature gradient, a flow of heat results in a direction opposite to the gradient. Thus if dT/dx represents the temperature gradient, the quantity of heat flowing per second is found from the expression. Q=K.A dT/dx If Q is expressed in watts, dT/dx in K per metre and the area of cross section A in sq.metres, then the coefficient of thermal conductivity, K is given in watts/metre x K. In insulating solids, the heat is carried by the lattice vibrations. This in part is also the case in metals, but the thermal conductivity due to the conduction electrons predominates in both insulators and conductors. The electrons in the hot end has a higher thermal energy. They move to the cold end where the excess energy is released to the atoms whereby the thermal agitation of the atoms and the temperature increase. The electrons of the cold end have less kinetic energy; so in passing to the hot end they decrease the thermal agitation and the temperature. Since the same electrons also conduct electric current, the transfer of heat and the conduction of current must be closely related processes. Finally, the total energy transferred across a cross-section is dependent upon 3 1. N, the number of electrons/m 2. V, the average velocity of the electrons 3. dW/dx, the energy gradient 4. λ, the mean free path 5. A, the area of cross-section or Q α NV.dW/dx x A x λ. Q=constant x N.V. x dW/dx.λA Further, since energy is a function of temperature which in turn is a function of position. Q=constant x N.V.λ dW/dT.dT/dx .A K=constant x NVλdW/dT . dW/dT is the rate at which the average energy of an electron increases with temperature. It is called the specific heat of an electron in the metal. THERMO ELECTRIC EFFECT : The basis of the study of thermo electric effects arises from the fact that electron motion is altered by the flow of current or by the application of temperature gradient. 1. Thomson Effect: If a piece of metal is made to have a temperature gradient between its two ends, an emf is observed to exist between those ends. This effect is known as Thomson effect, arise since electrons at the hot end tend to move to the cold end. A space charge is established in the metal producing an electric field the direction of which is from the hot end tend to move to the cold end. This electric field tends to drive the electrons from the cold end to hot end. When equilibrium is reached the two effects is cancelled. Under these condition the electric field is proportional to the temperature gradient. The temperature gradient is negative, because as the distance from the hot end increase, the temperature decreases. Thus electric field is in opposition to the temperature gradient. E=− Where is the temperature gradient, and is the thomoson coefficient which is expressed in units of volts/℃. 2. Seeback Effect : The thermocouple was discovered by seeback in1822 when he demonstrated that a loop composed of two dissimilar metals could be made to carry a continuous current simply by maintaining the two junctions at different temperatures. The magnitude of the current depends on the resistance of the metals. When the two metals are placed in contact, then a contact potential equal to the difference in work function of the two metals is established at the junction. The work function is defined as the difference between the escape level and the Fermi level. The Fermi level is subjected to a small temperature change of order -5 -4 of 10 - 10 eV/K. Thiscauses a difference in the contact potentials at the two junctions due to the different temperatures at the two ends and the results is an emf which is free to drive the current. 3. Peltier Effect : In 1834, Peltier discovered the converse effect and the showed that when a current is passed through the junction of two different metals, heat is absorbed or liberated depending on the direction of the current. Thus if the Seebackemf is from metal A to metal B at the hot junction, an external emf applied in this direction will produce a cooling effect at this junction. The heat is referred to as “Peltier heat” and it is equal to the work done in transferring a charge q from metal A to metal B or π joules, where π A-B is the peltier coefficient. CHAPTER - II INTRODUCTION Insulators or dielectrics as distinct from conductors have no free electrons. Hence when a source of e.m.f is connected across a dielectric no current flows. However, since no dielectric is perfect it contains a small number of free electrons and a very small current flows through it when an electric field is applied. Capacitors therefore have a small leakage conductance. EFFECT OF A DIELECTRIC ON THE BEHAVIOUR OF A CAPACITOR Suppose that two large plane parallel plates separated by a distance d (meters) in vacuum are maintained at a potential difference V. The plates will become charged positively and negatively with charges = Q and a uniform electric field intensity E = V/d (volts/m) will be o created between the plates. The magnitude of the charge accumulated on each plate is proportional to the applied potential difference, i.eQ∝ V or Q = C V, where C is defined as o o o o the capacitance. By applying Gauss theorem, the magnitude of flux density D within the plates is given by Since the electric field strength E is related to the flux density by the relation D = E, the o field strength in the region between the plates is given by E = D/ = Q /A Since V = Ed, o o o. the capacitance of the system is given by C = A/d where is termed as the absolute o o o permittivity of free space. If the space between the plates is now filled with a dielectric and V is kept constant, it is found that the value of the charge is increased to Q = CV. It follows that the new capacitance is given by C = A/d where is defined as the absolute permittivity of the dielectric and the ratio = C/C =/ is called the relative permittivity, r o o specific inductive capacity or the dielectric constant of the material. The dielectric constant of a medium is constant if the state of the medium doesn’t vary from point to point. At the boundary between two media the dielectric constant changes abruptly, and bodies that are non-homogeneous with respect to density and other properties are usually non-homogeneous with respect to the dielectric constant. POLARISATION A dielectric consists of molecules the atomic nuclei of which are effectively fixed, relative to each other. In absence of any external field the electrons are distributed symmetrically round the nucleus at any instant. When an electric field is applied the electrons of the atoms are acted upon by this field. This causes a movement of the electrons which are displaced in a direction opposite to that of the field. The resultant effect is to separate the positive and negative charges in each molecule so that they behave like electric dipoles. The strength of each dipole is given by the dipole moment which in its simplest form consists of two point charges of opposite sign ±Q separated by a distance d. The dipole moment has magnitude Qd and is represented by a vector pointing from the negative charge in the direction of the positive charge. The dipole moments are expressed in terms of the Debye unit. When the dipoles are created the dielectric is said to be polarized or in state of polarization .When the field is removed and the atoms return to their normal or unpolarised state, the dipole disappear. The polarized dielectric consists of a layer of dipole as shown in fig below There is an induced negative charge on the surface of the dielectric near the positive plate and a similar induced positive charge on the surface near negative plate. There is no resultant charge density at any point within the dielectric because all individual dipole are aligned parallel to the field, each negative charge of the one dipole being next to the positive charge of the next dipole. Consider the dielectric to be composed of a large number of elementary cylinder each of length l in the direction of the applied field and of cross section δA. Let a uniform field of strength E be applied normal to the plates .This polarizes the dielectric inducing dipoles in each elementary cylinder and charges δq appear on either end of the cylinder. The charge density, σ on the surface δA of the cylinder given by σ = δq/δA =l.δq/l.δA =m/δV Where m is the dipole moment and δV is the volume of the elementary cylinder. If the number of dipoles per unit volume be N i.e., if N=I/Δv; then σ =Nm. The product Nm is called the polarization (P) of the dielectric and is the total dipole moment established within unit volume of the insulating medium. Thus a dielectric subjected to a homogenous field carries a dipole moment P per unit volume which may be written as P=Nm The charge density σ is a scalar quantity but the polarization P is a vector quantity because it involves direction. For any dielectric, σ is equal to the normal component of the polarization. For an isotropic dielectric, the direction of polarization is perpendicular to the plates. Hence we may write σ =P , where P is the component of polarization perpendicular to the plates. n n The electric polarization of a dielectric maybe conceived as a forced state of the medium caused by the action of an electromotive force and which disappear when that force is removed. In other words, it is a displacement of charge produced by an electromotive intensity. When emf acts on a conducting medium it produces a current through it, but if the medium is a non-conductor or dielectric, the current cannot continues so flow through the medium but electric charge would be displaced within the medium in the direction of the electromotive intensity. If σ represent the charge density on the plates of a condenser containing no dielectric and if 0 σ represent the charge density on the plates of the condenser filled with a homogenous 1 dielectric then σ =Q /A=C V/d= V/d= E 0 0 0 0 0 n Where E is the normal component of the electric field strength, n Similarly σ =Q/A=C V/d= V/d=E 1 n i.e there is an increase in charge density. The increase may be observed experimentally. On removal of the dielectric, this additional charge return to the source. The additional attraction of charge to the condenser plates is explained by assuming that charges of opposite sign having a density E ( -1) are formed on the surface of the dielectric next to the condenser 0 n n plates, Thus we may write, σ σ 1 0 Where σ is the charge density due to polarization or E = E + P Where P = E ( -1). pol n 0 n n n 0 r The above expression may be written in the generalized from as, E = E+P 0 P=(- )E 0 = E( -1) 0 1 Or P E, stating that the polarization P of a substance is proportional in magnitude to the σ applied field E, at all ordinary field strength, provided that the dielectric constant is r independent of the applied field which it is for normal dielectrics below the breakdown field. 2 The magnitude of polarization is expressed in coulombs/m . Since polarization P is proportional to the dipole moment m, the latter must be proportional to the electric field strength or m=αE.α is proportionality constant and is called the polarizability of the elementary dipole volume. In deriving P= E ( -1) the physical state of the dielectric was no considered. 0 1 3 Consider a gas containing N atoms/m subjected to a homogenous field E, Neglecting any interaction between the dipoles induced in the atoms, which is an odd approximation for a gas , we find for the polarization of a gas , P=Nm=NαE Comparing this expression with the macroscopic expression for P, We have =1+Nα/ r 0 When two plates in vacuum are charged initially to potential V, a uniform field is created between them and the intensity of the field is given by V/d. When a slab of permittivity is interposed between the plates, the capacity of the system is reduced by the factor . But since the potential difference between the plates is maintained by the battery, the charge on the plates should increase by a factor .H ence the charge per unit area is given by D = E n 0 In accordance with the ideas of displacement a further quantity of charge in displaced per unit area between the plates. The additional charge displace per unit area at any point is represented by the normal component of the polarization vector P. Hence, D = E +P n 0 n n In general, the total displacement D at any point is now given by D = D + P 0 Or D = E+ p 0 The above equation is valid for even anisotropic media where the polarization vector is not necessarily parallel to the electric field vector. Since P =NαE D =E ( + N ) 0 α = + ( – 1) 0 r 0 = 1 + ( N/ ) r 0 D= E 0 r The above equation is valid only for isotropic material where the permittivity , remains r constant in all directions. In crystals generally depends on the direction along which it is r, measured relatively to the crystal axes. In polycrystalline materials, on the other hand, with a random distribution of gains, the directional effects disappear. The essence of all electrostatic problems in the presence of dielectric materials is the determination of polarization P. All dielectric application depends upon the ability to vary P in some manner. P may be varied by changing the electric field, temperature, or mechanical strain. In most problems, it is required to find out the manner in which P varies with the electric field E. In an anisotropic material, the relationship between P and E may be very complex because the resultant polarization in a given direction may be a function of electric fields in all three mutually perpendicular directions. The simplest case is the three P is directly proportional to E i.e P =K E. K is a dimensionless scalar quantity and is defined as 0 the dielectric susceptance of the medium. Under these conditions, = 1 +K. r In case of isotropic materials, the relation between P and E may exhibit hysteresis in which case D≠ E. Such materials are non-linear and are called ferroelectric materials. The 0 r figure below shows the polarization curve for such a material. Here it is possible to define a number of dielectric constants. The initial dielectric constant may be defined as the slope of the normal polarization curve at E =0. The incremental dielectric constant is defined as the limit of In ferroelectric materials the electric flux lags behind the electric force producing it such that under varying electric forces a dissipation of energy occurs. The energy is dissipated as heat. The energy loss due to this case is called the dielectric hysteresis loss. The dissipation of energy may be explained by assuming a continual charge in the orbital paths of the electron in the atomic structure due to a varying or alternating electric stress in the dielectric. Dielectric hysteresis, however, cannot be measured as a separate quantity and in practice the total dielectric losses (including losses due to a small conduction current) are usually measured by means of an a.c bridge. THE DIELECTRIC CONSTANT OF MONOATOMIC GASES In a gas the average distance between the atoms or molecules is large enough so that one neglect interaction between them and the individual atom can be studied independently. Consider a single atom consisting of a positive nucleus of charge Ze with Z electrons moving around the nucleus. Let us assume that the total negative charge –Ze is distributed homogeneously throughout a sphere of radius “a”. When this atomic model is placed in a field E as shown in figure above, there is no translational force on the atoms as a whole, since it is electrically neutral. But the nucleus and the electron cloud will evidently try to move in opposite direction because of opposite signs of their charges. However as they are pulled apart, a force will develop between them tending to bring the nucleus back to the centre of the sphere. Consequently, an equilibrium condition will be obtained in which the nucleus if displaced relative to the centre of the electron cloud in the direction of E. the displacement of nucleus may be calculated as follows: assume that the nucleus is shifted by an amount “d” as shown in the figure. The force on the nucleus in the direction of the field equals ZeE. The electron cloud can be divided into two regions- one inside an imaginary sphere of radius “d” and other between the two spherical surfaces of radii “d” and “a”. From Gauss’s theorem the charge in the latter region does not exert any force on the nucleus. The only force exerted on the nucleus is the one produced by the negative charge which is distributed inside the smaller sphere of radius “d”. The charge inside this sphere is equals to The force exerted on the nucleus by this charge assuming it to be concentrated in the centre of the sphere is given by Coulomb’s law. In equilibrium condition, one obtains Hence, the displacement of the nucleus relative to the centre of the sphere is given by This shows that the displacement is proportional to the field strength- a situation akin to the one is in which a mechanical force is exerted on a particle bound with an elastic force to a certain equilibrium position. On the application of an electric field the atom still remains neutral but has a non-zero dipole moment due to displacement of nucleus relative to the centre of the electron cloud. The magnitude of the dipole moment is given by 3 m= Ze.d= 4π a E 0 This expression shows that the dipole moment is proportional to the volume of the electron 3 cloud. The polarizability α is thus equation to 4π a . 0 Finally using the equation the dielectric constant of a monoatomic gas may be expressed as 3 = 1 + 4πa N r

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