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Note for Electomagnetic Waves and Transmission Lines - EMTL by Pooja Honey

  • Electomagnetic Waves and Transmission Lines - EMTL
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  • Electronics and Communication Engineering
  • B.Tech
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www.alljntuworld.in JNTU World or ld blocks of matters. Quarks were predicted to carry a fraction of electronic charge and the existence of Quarks have been experimentally verified.] Principle of conservation of charge states that the total charge (algebraic sum of positive and negative charges) of an isolated system remains unchanged, though the charges may redistribute under the influence of electric field. Kirchhoff's Current Law (KCL) is an assertion of the conservative property of charges under the implicit assumption that there is no accumulation of charge at the junction. Electromagnetic theory deals directly with the electric and magnetic field vectors where as circuit theory deals with the voltages and currents. Voltages and currents are integrated effects of electric and magnetic fields respectively. Electromagnetic field problems involve three space variables along with the time variable and hence the solution tends to become correspondingly complex. Vector analysis is a mathematical tool with which electromagnetic concepts are more conveniently expressed and best comprehended. Since use of vector analysis in the study of electromagnetic field theory results in real economy of time and thought, we first introduce the concept of vector analysis. W Vector Analysis: TU The quantities that we deal in electromagnetic theory may be either scalar or vectors [There are other class of physical quantities called Tensors: where magnitude and direction vary with co ordinate axes]. Scalars are quantities characterized by magnitude only and algebraic sign. A quantity that has direction as well as magnitude is called a vector. Both scalar and vector quantities are function of time and position . A field is a function that specifies a particular quantity everywhere in a region. Depending upon the nature of the quantity under consideration, the field may be a vector or a scalar field. Example of scalar field is the electric potential in a region while electric or magnetic fields at any point is the example of vector field. A vector can be written as, , where, is the magnitude and unit vector which has unit magnitude and same direction as that of and JN Two vector are added together to give another vector is the . . We have ................(1.1) Let us see the animations in the next pages for the addition of two vectors, which has two rules: 1: Parallelogram law and 2: Head & tail rule JNTU World

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www.alljntuworld.in JNTU World Commutative Law..........................................(1.3) Associative Law.............................................(1.4) The position vector of a point P is the directed distance from the origin (O) to P, i.e., . W = or ld Distributive Law ............................................(1.5) Fig 1.3: Distance Vector = OQ are the position vectors of the points P and Q then the distance TU If = OP and vector Scaling of a vector is defined as , where is scaled version of vector scalar. Some important laws of vector algebra are: and is a JN Product of Vectors When two vectors and are multiplied, the result is either a scalar or a vector depending how the two vectors were multiplied. The two types of vector multiplication are: Scalar product (or dot product) Vector product (or cross product) gives a scalar. gives a vector. The dot product between two vectors is defined as Vector product JNTU World = |A||B|cosθAB ..................(1.6)

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www.alljntuworld.in JNTU World and or ld is unit vector perpendicular to Fig Product of Vectors W When two vectors and are multiplied, the result is either a scalar or a vector depending how the two vectors were multiplied. The two types of vector multiplication are: Scalar product (or dot product) gives a scalar. Vector product (or cross product) gives a vector. The dot product between two vectors is defined as = |A||B|cosθAB ..................(1.6) Vector product and JN TU is unit vector perpendicular to Fig Fig 1.5 :Illustrating the left thumb rule for determining the vector cross product JNTU World

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www.alljntuworld.in JNTU World The dot product is commutative i.e., and distributive i.e., . Associative law does not apply to scalar product. The vector or cross product of two vectors and is denoted by . is a vector or ld perpendicular to the plane containing and , the magnitude is given by and direction is given by right hand rule as explained in Figure 1.5. ..................................................... .......................................(1.7) where is the unit vector given by, . The following relations hold for vector product. i.e., cross product is non commutative ..........(1.8) W = i.e., cross product is distributive.......................(1.9) i.e., cross product is non associative..............(1.10) TU Scalar and vector triple product : Scalar triple product .................................(1.11) Vector triple product ...................................(1.12) Co-ordinate Systems JN In order to describe the spatial variations of the quantities, we require using appropriate co-ordinate system. A point or vector can be represented in a curvilinear coordinate system that may be orthogonal or non-orthogonal . An orthogonal system is one in which the co-ordinates are mutually perpendicular. Nonorthogonal co-ordinate systems are also possible, but their usage is very limited in practice . Let u = constant, v = constant and w = constant represent surfaces in a coordinate system, the surfaces may be curved surfaces in general. Furthur, let JNTU World , and be the unit

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