Discipline Course-I. Semester-II. Paper No: Electricity and Magnetism. Lesson: Polarization and Dielectric Materials

Size: px

Start display at page:

Download "Discipline Course-I. Semester-II. Paper No: Electricity and Magnetism. Lesson: Polarization and Dielectric Materials"

Theodore Griffin
5 years ago
Views:

1 Discipline CourseI SemesterII Paper No: Electricity and Magnetism Lesson: Polarization and Dielectric Materials Lesson Developer: Dr. Amit Choudhary College/ Department: Physics Department, Deshbandhu College, Kalkaji, University of Delhi 1

2 Table of Contents Title: Polarization Introduction Different types of polarization Electronic polarization(or atomic polarization) Ionic polarization Orientation polarization Electric field outside the dielectric body Electric field inside a dielectric Relation between E, P, and D fields Relative dielectric constant Electric displacement Electric susceptibility Gauss law in dielectrics Questions Multiple choice questions References 2

3 Learning Objective This lesson aims at the following student learning objectives. Meaning of polarization of a dielectric and types of Polarization Mechanism of polarization of nonpolar dielectrics. Concept of the polarization vector, P. Concept of polarization charges Effects of polarization of a dielectric on the net electric field inside a dielectric Know the meaning of dielectric constant Know the meaning of, and definition of, the term electric Susceptibility Know the reason for thinking in terms of another vector field, instead of the usual electric field, in the presence of dielectrics Define the displacement vector a) Understand the significance of D b) Know the relation between the vectors E, P c) Know the modified form of Gauss s law in dielectrics 13.1 Introduction As we have discussed in the chapter based on dielectric materials that in the presence of electric field the center of positively charged nucleus and negatively charged electrons of molecules of a dielectric materials get separated when electric field is applied resulting in the creation of dipole moments in the direction of applied electric field. For example, consider an atom, Fig 1(a), in the absence of external field. It shows that the center of mass of positive and negative charges coincides at the center of the atom. Now, when the external electric field is applied, the centers of mass charges get separated Fig. 1(b). If the distance between two center of masses of opposite charges is and the net magnitude of each type of charges is, then the net dipole moment is. If the dipole moments in the dielectric material are generated only by the application of electric field, then they are called induced dipole moments. However, some dielectric materials possess dipole moments in the absence of external electric field also, eg. H 2 O, HCl etc., then they are called as permanent dipole moments. 3

4 (a) + (b) + (c) + q q Fig. 1. (a) Schematic of atom in neutral state i.e. no external field is applied, (b) external electric field is applied on the atom, and (c) the center of mass get separated with a distance. If the dielectric material possesses randomly oriented permanent dipoles, the application of external electric field tends to align them along the direction of applied electric field. The materials in such state of creation of induced dipoles or aligning the permanent dipole are called polarized materials and the phenomenon is called electric polarization. The polarization field in the dielectric is therefore opposite to the direction of external applied field. However, the polarization density is in the direction of applied electric field. The polarized material produces its own electric field, both at exterior points and inside the dielectric as well. The polarization of dielectric depends on the net electric field in it. The field just outside the polarized medium can modify the free charge distribution of electron in the conductor placed close to it which in turn can change the electric field in the dielectric. Mathematically, this can be derived by assuming a small volume element v of a dielectric medium which is as a whole electrically neutral. On the application of electric field, the dielectric medium becomes polarized. Then, the volume element is characterized by the dipole moment as (1) Where is the position vector where the electric field is to be observed. is the charge element which has the relation with volume in terms of volume charge density as. This quantity determines the electric field produced by the small volume at a point outside the volume element. As we have seen that depends on the size of volume element, therefore it would be more easy to work with dipole moment per unit volume, Fig. 2, so that it could be observed independent of volume of dielectric, 4

5 (2) The quantity is called as electric polarization or simply polarization as it becomes point function when is limited to a very small volume. So, is a vector quantity in the direction of. has the direction of displacement of separation of positive and negative charges in atoms or molecules. The macroscopic small volume element still has large numbers of molecules which are contributing to the p. Thus assuming the contribution of individual molecule in polarization, (3) P Figure 2 The schematic of electric polarization. The net dipole moment due to is, where summation is extended to all the molecules inside volume element. Therefore, we can write (4) The net polarization is the dipole moment per unit volume of the dielectric material as given by Eq Different types of polarization: There are three types of electric polarization Electronic polarization (or atomic polarization): An atom consists of positively charged nucleus surrounded by negatively charged electron cloud as shown in Fig. 1(a). The center of masses of positive and negative charges coincide in the absence of external electric field. In this state, there is no electric polarization. When external electric field is applied, the center of positive and negative charges get distorted form their positions and induces a displacement up to a certain distance in the direction of applied electric field between the center of positive and negative charges as shown in Fig. 1(b). The positive charge shifts in the direction of electric field whereas the negative charges shift opposite to this. Thus, a dipole is induced as discussed in the above section as discussed in introduction part of this chapter. The induced dipole is proportional to the applied electric field (5) 5

The constant of proportionality is called the atomic polarizability. For animations ctrl+click on the following link: http://www.doitpoms.ac.uk/tlplib/dielectrics/polarisation_mechanisms.php 13.2.

6 The constant of proportionality is called the atomic polarizability. For animations ctrl+click on the following link: Ionic polarization: Let us discuss about ionic polarization. First consider two ions (cation and anion) which are connected by ionic bond. The ionic bond has certain amount of energy i.e. the minimum energy required to break the bond which is called as threshold energy. If we apply the electric field less than the threshold field then the bond acts as a spring. Therefore the ions associated with this bond would stretch and contract on the application of external electric field i.e. the ions oscillate about their equilibrium position. This stretching of ions from their equilibrium results in the separation of center of masses of positive and negative charges and generates the dipole moment. The polarization so produced is called as ionic polarization. 6

7 polarization.swf Fig. 3. (a) Ions (anion and cation) are attached with ionic bond in the absence of external electric field, (b) Ions get stretched on the application of electric field. Examples of ionic polarization are NaCl, CaCl 2, BaO etc. For animation ctrl+click on the following link: 7

8 O O O O Polarization and Dielectric materials Orientational polarization: Consider a material in which the molecules possess permanent dipole moment,. In a piece of bulk material, all the dipoles of molecules are oriented in random directions in the absence of an electric field. The randomly oriented dipoles give net zero polarization (Fig. 4 (a)),. This is due to the fact that the effect of each dipole is cancelled by the other dipole. But if an electric field is applied in a direction, all the dipoles rotate along the direction of electric field. For example water (H 2 O), the dipole moment of H 2 O is 1.85 C.m or 1.85 D ( D stands for Debye. This is a unit, which is the measure of dipole moment). (a) (b) O O O O O O O O O O =0 >0 Fig. 4 (a) The net polarization is zero in the absence of electric field, (b) dipoles are oriented in the direction of applied electric field. 8

9 For more animations ctrl+click on the link below: 9

10 Value addition: Did you Know The properties of material changes with dipole moment. The variation in the dipole moment brings the change in the intrinsic properties of material. For example water, the dipole moment of water is about 1.85 D whereas the dipole moment of crystal water i.e. ice is 3.1 D. The bond angle between hydrogen atoms of ice at room temperature is 105 o whereas this is o in ice form. This happens because there are six valance electrons in oxygen and one electron is contributed by hydrogen atom in H 2 O. That means total eight electrons from two HO bonds. In this process two lone pairs electrons are left unpaired. In equilibrium, the lone pairs and bonds stay away from each other. Then they are repelled by each other to stay away in equilibrium forming a tetrahedral structure. Such an ideal structure should give HOH bond angle of O The application part of water dipole moment can be seen in the current technology of microwave oven. The dipole moment of water tries to align in the direction of alternating electric field of microwave radiations passing through the food items. The alternating dipoles rotate the water molecules and heats up the nearby molecules linked to our food item as a result of this, the food items gain heat energy and becomes hot. Therefore, this concept is used in heating the food items containing water like contents. Source: ates_of_matter/hydrogenbonding_and_water HAC&pg=PA200&lpg=PA200&dq=dipole+moment+of+water+h2O+ic e&source=bl&ots=lbsdyiyaud&sig=j ui6tn3gfklatqxuc5g7swkigk&hl=en&sa=x&ei=dkqyutw 10

EMP_rQftuIFI&ved=0CEgQ6AEwBA#v=onepage&q=dipole%20momen t%20of%20water%20h2o%20ice&f=false http://en.wikipedia.org/wiki/microwave_oven 13.

11 EMP_rQftuIFI&ved=0CEgQ6AEwBA#v=onepage&q=dipole%20momen t%20of%20water%20h2o%20ice&f=false Electric field outside the dielectric body: We consider a polarized dielectric body. Assume a small volume element at position. Its polarization at a point is. This gives rise to the electric field outside the dielectric material. Now, in order to calculate the electric field outside, we need to calculate the potential at that point negative gradient of which will give the electric field. Consider the small volume of dielectric with resulting dipole moment p. Now we have to calculate the potential at a point distance from the position of volume element at. This distance is large as compared to the dimension of volume element. Then the expression for the potential is Here (6) Fig. 5. The electric field at may be calculated by summing up the caontribution due to the various volume elements in. The surface of is denoted by. For the calculation of net potential due to the contribution from each volume element, we have to integrate the above expression (7) Now mathematically we can write 11

12 The operator By using a derivative identity, will operate over the primed quantities. Therefore, (8) (9) Using the above identity in Eq. 9, we can write Eq. 8 as (10) The final potential can be written in modified way by using above expression and Gauss s divergence theorem for first part on RHS in above expression, (11) Where is the unit vector over the surface of cross section of volume. The quantities and in the above expression are the two quantities obtained from polarization P. The dimension of these quantities are charge per unit area and charge per unit volume. So, they can be defined as (12) (13) Thus Eq. 12 and 13 can be called as polarization charge densities. and are the surface and volume charge densities respectively Electric field inside a dielectric Now one has to think about the clear meaning of electric field inside the dielectric. Basically, we are interested to know the average macroscopic field inside the small volume containing large number of molecules which are contributing to the polarization of dielectric material. From fundamental point of view, the definition of macroscopic electric field inside the dielectric can be given as the electric field is the force per unit charge on a test charge embedded in the dielectric, in the limit where the test charge is so small that it does not affect the charge distribution. Such a dimensionally small charge can be called as point charge when we are dealing with the macroscopic observation of electric field. As the calculation of macroscopic electric field is difficult by assuming taking the direct consideration of above definition, therefore an indirect approach of the calculation of electric potential has been applied. Once the electric potential is known the electric field can be calculated without difficulty. The electrostatic field in a dielectric must have the same basic properties that we found applied to. is a conservative field and can be derived from electric potential. Also we know that the curl of is zero. Then surface integral of above equation can be written as Applying Stoke s theorem on the above expression for the conversion of surface integral to 12

13 line integral, we get (14) In order to explain the integral, let us consider the needle shaped volume cavity as shown in the Figure 3 below Dielectric material S 1 A B S 2 D C Fig. 6. The schematic of the needle shaped cavity in the dielectric material with path ABCD. Applying the above equation to the path ABCD, the integral will be calculated over the path AB and CD only as the path BC and DA are negligibly small. The integral becomes (15) Where and and stands for vacuum and dielectric respectively. stands for the tangential component. (16) Consequently, the electric field inside the dielectric is equal to the electric field inside the needle shaped cavity in dielectric only if the cavity is oriented along the direction of field. But the cavity is nothing but the area outside the dielectric. Hence, the calculation of electric field at point would provide the electric field at the point which would be valid for inside the dielectric. Therefore, one can use the scalar potential derivation as done in above heading (17) Where is the volume excluding the needle shaped cavity. is the exterior surface and S =S 1 +S 2 +S c is the surface of the dielectric. S 1,S 2, and S c are the surface area of cross section 1 and 2 whereas S c is the area of curved surface. 13

14 13.5 Relation between three electric vectors, and Following are the points which need to be discussed to establish the relationship among, and : Relative dielectric constant Consider a parallel plate capacitor filled with dielectric material. The bound charges +q and q will be on the interface of the plates of capacitor. These charges are opposite to the real charges on the plates and tend to decrease the electric field in between the parallel plates of the capacitor. Thus, the net electric field within the dielectric material would be less than the electric field within the capacitor without the dielectric medium. Thus, the ratio of electric field between parallel plates of capacitor when air is filled to that the electric field when dielectric material is filled between plates and can be represented as. (18) The quantity is called as the relative dielectric constant or relative permittivity. Where is the electric field in air and is the electric field in dielectric. and are the permittivity of free space and dielectric medium respectively. is the distance between parallel plates of capacitor. Substituting these values in above Eq. (19) Electric displacement vector As we have seen in the above topic that the electric field between the parallel plates of a capacitor depends up on the dielectric medium. A new electric field vector can be considered to explain this effect. The new considered electric vector is electric displacement,. This depends on the distribution and magnitude of the charges which produce electric field but it is independent on the nature of the medium. Now consider a single point charge, distance is and electric displacement due to it at a (20) This is very clear that is dielectric constant dependent and has the same direction as in case of isotropic mediums Electric susceptibility We have seen that the dielectric materials give response to the electric field. The susceptibility of dielectric material indicates the extent to which the dielectric material is polarized i.e. the degree of polarization of a dielectric material. The polarization is the function of electric field i.e. (21) We know the total charge density,,where and are the bound and free charges within the dielectric material. is the susceptibility of dielectric material. Now, we know the Gauss s law leads to (22) Substituting the expression of volume bound charge density from Eq

15 (23) Thus, the displacement electric field D is (24) This is the relationship between three vectors displacement field and polarization. Now, substituting the expression of P in the above expression., electric (25) Where, is the dielectric constant of the medium. Thus, from the above relation (26) Thus Eq. (26) represents the relation between susceptibility and relative dielectric constant Gauss s law in dielectric medium In dielectric materials, the electric field is maintained by polarization of charges inside the material itself which is not applicable to vacuum where no charge is available for the same. In order to know the electrostatic field inside the dielectric, the Gauss s law can be employed which shows the that the electric flux across any arbitrary closed surface is directly proportional to the net charge enclosed by the surface. For the implementation of Gauss s law, we need to consider all types of charges responsible for the generation of net electric field. For this purpose, we consider the polarization charge ( ) and some extra charge ( ) embedded externally (composition of some charges,,,, etc.). The Gauss s law for dielectric materials can be expressed in two forms, integral and differential. By the definition of Gauss s law, we have (27) where,, and are the electric field, unit vector to the surface. is the permittivity of free space. The polarization charge can be expressed as (28) Then, substituting the expression of surface and volume charge densities from Eqs (12) and (13), we get (29) If we consider the embedded charges, and bounded by surfaces,, and respectively which has the contribution to the net surface enclosing these charges as shown in Fig. 4. The dielectric material has no boundary at surface which shows that there will not be any contribution from boundary of material. 15

16 Now converting the volume integral in to surface integral by the application of Gauss divergence theorem, the first term will be cancelled by the second on left hand side of Eq. (29). Then, we get (30) Substituting Eq. 30 in to Eq. 27, we get (31) S 2 q 2 S 3 S 1 q 1 q 3 Dielectric Figure 4. The dielectric medium embedded with charges, and enclosed by their respective surfaces and the system of these charges is enclosed by a surface S. Substituting Eq. (27) in Eq. (31), This is the integral form of Gauss s law for the electric displacement in dielectrics. If this is applied to a small volume element v then it becomes as Dividing the above relation by v and proceeding to the limit, we get This relation is the differential form of Gauss s law Questions Multiple choice questions Question Number Type of question 1 Yes/No 1.1. The units of displacement field and polarization are different. Yes/No 1.2. The electric field produced by polarization inside the polarized dielectric is 16

17 opposite to the applied electric field. Yes/No The electric susceptibility is not a unitless quantity. Yes/No 1.4. The external charge on the surface of polarized dielectric is free. Yes/No 1.5. The polarization charge inside the dielectric is also called bound charge. Yes/No Answers 1.1 No 1.2 Yes 1.3 No 1.4 No 1.5 Yes Question Number Type of question 2 Multiple choice 2.1. The unit of dipole moment. A. C/m B. Cm C. Cm 2 D. C/m The unit of polarization in a dielectric is A. C/m 2 B. Cm C. Cm 2 D. C/m If the electric susceptibility of a dielectric is 7, its relative permittivity is A. 8 B. 9 C. 6 D In vacuum P=0, therefore E can be written as A. o D B. D C. D. D/ o 2.5. If the polarization of a dielectric is parallel to the surface of it, then the surface charge density of a polarized dielectric is 17

18 A. Zero B. P C. P D..P Answers 2.1 B 2.2 C 2.3 A 2.4 D 2.5 A Subjective questions: 1. What do you understand by electric polarization? How many types of electric polarization are there? 2. Derive the expression of electric field outside the dielectric polarization. 3. Derive the relation between the three electric vectors, and. References 1. J. R. Reitz, F. J. Milford, and R. W. Cristy, Foundations of electromagnetics theory,narosa Publishing House, New Delhi. 2. David J. Griffiths, Introduction to Electrodynamics, 3rd edition. 3. S. L. Kakani and A. Kakani, Material Science, New Age International Publishers, New Delhi. 18

Chapter 4. Electrostatic Fields in Matter

Chapter 4. Electrostatic Fields in Matter 4.1. Polarization 4.2. The Field of a Polarized Object 4.3. The Electric Displacement 4.4. Linear Dielectrics 4.5. Energy in dielectric systems 4.6. Forces on