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Note for ELECTROMAGNETIC THEORY AND TRANSMISSION LINE - ETTL by Ramanjaneya Reddy G

  • ELECTROMAGNETIC THEORY AND TRANSMISSION LINE - ETTL
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  • Electronics and Communication Engineering
  • B.Tech
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Introduction In the previous chapter we have covered the essential mathematical tools needed to study EM fields. We have already mentioned in the previous chapter that electric charge is a fundamental property of matter and charge exist in integral multiple of electronic charge. Electrostatics can be defined as the study of electric charges at rest. Electric fields have their sources in electric charges. ( Note: Almost all real electric fields vary to some extent with time. However, for many problems, the field variation is slow and the field may be considered as static. For some other cases spatial distribution is nearly same as for the static case even though the actual field may vary with time. Such cases are termed as quasi-static.) In this chapter we first study two fundamental laws governing the electrostatic fields, viz, (1) Coulomb's Law and (2) Gauss's Law. Both these law have experimental basis. Coulomb's law is applicable in finding electric field due to any charge distribution, Gauss's law is easier to use when the distribution is symmetrical. Coulomb's Law Coulomb's Law states that the force between two point charges Q1and Q2 is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Point charge is a hypothetical charge located at a single point in space. It is an idealized model of a particle having an electric charge. Mathematically, , where k is the proportionality constant. In SI units, Q1 and Q2 are expressed in Coulombs(C) and R is in meters. Force F is in Newtons (N) and , is called the permittivity of free space. (We are assuming the charges are in free space. If the charges are any other dielectric medium, we will use instead where dielectric constant of the medium). is called the relative permittivity or the

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Therefore ....................... (1) As shown in the Figure 1 let the position vectors of the point charges Q1and Q2 are given by and . Let represent the force on Q1 due to charge Q2. Fig 1: Coulomb's Law . We define the unit vectors as The charges are separated by a distance of ..................................(2) and . can be defined as Similarly the force on Q1 due to charge Q2 can be calculated and if represents this force then we can write When we have a number of point charges, to determine the force on a particular charge due to all other charges, we apply principle of superposition. If we have N number of charges Q1,Q2,.........QN located respectively at the points represented by the position vectors , ,...... , the force experienced by a charge Q located at is given by,

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.................................(3) Electric Field : The electric field intensity or the electric field strength at a point is defined as the force per unit charge. That is or, .......................................(4) The electric field intensity E at a point r (observation point) due a point charge Q located at (source point) is given by: ..........................................(5) For a collection of N point charges Q1 ,Q2 ,.........QN located at field intensity at point , ,...... , the electric is obtained as ........................................(6) The expression (6) can be modified suitably to compute the electric filed due to a continuous distribution of charges. In figure 2 we consider a continuous volume distribution of charge (t) in the region denoted as the source region. For an elementary charge , i.e. considering this charge as point charge, we can write the field expression as: .............(7)

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Fig 2: Continuous Volume Distribution of Charge When this expression is integrated over the source region, we get the electric field at the point P due to this distribution of charges. Thus the expression for the electric field at P can be written as: ..........................................(8) Similar technique can be adopted when the charge distribution is in the form of a line charge density or a surface charge density. ........................................(9) ........................................(10) Electric flux density: As stated earlier electric field intensity or simply ‘Electric field' gives the strength of the field at a particular point. The electric field depends on the material media in which the field is being considered. The flux density vector is defined to be independent of the material media (as we'll see that it relates to the charge that is producing it).For a linear isotropic medium under consideration; the flux density vector is defined as: ................................................(11) We define the electric flux as

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