Electromagnetic Field and Wave Theory Assignment Backup

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<p>NAME OF STUDENT: EMERIBE CHIKE IKWUDIMMA</p> <p>STUDENT NUMBER: UAD14303SEL21755</p> <p>DEPARTMENT: ELECTRICAL ENGINEERING DEPARTMENT</p> <p>DEGREE: MASTERS/DOCTORATE</p> <p>COURSE TITLE: ELECTROMAGNETIC FIELDS &amp; WAVES THEORY</p> <p>ASSIGNMENT PAPER</p> <p>1.0.</p> <p>INTRODUCTION</p> <p>The field of electrical engineering can be broadly classified into two namely the electric circuit theory and electromagnetic fields and wave theory. While circuit theory deals with electrical analysis, synthesis and design of a wired networks, electromagnetic theory focuses on the propagation of electrical signals through a wireless medium. This assignment takes an in depth look at some concepts in electromagnetic theory as well as its applications. In the field of electromagnetic theory, one name stands very tall, the man whom many have described as the wizard of electromagnetic theory. He is James Clerk Maxwell. In 1863, James Clerk Maxwell before the Royal Academy of science proved that electrical signals can be propagated through a wireless medium in form of electric and magnetic fields. This he proved possible through the introduction of a displacement current concept. Years latter, this theory of Maxwell was practically verified and proved correct by the likes of Guglielmo Marconi and Hertz. James Clerk Maxwell developed four basic equations that characterized both electric and magnetic fields, these equations till date form the basis for the study and analysis of electromagnetic fields and wave theory.</p> <p>1.1.</p> <p>THE CONCEPT OF ELECTRODYNAMICS Electrodynamics deals with electric and magnetic fields, its inter-</p> <p>relationships under various charge arrangements. Electrodynamics can be classified into two areas viz: Electrostatics and Magneto statics ELECTROSTATICS: This deals with theory that describes physical phenomena related to the interaction between stationary electric charges or charge distributions in space. Recall from coulombs law, the force on a charge q located at a distance r due to the existence of another charge relation located at a distance is given by the</p> <p>The interpretation of the above equation(1) is that the electrical force between two charges separated by distance R is directly proportional to the product of the magnitude of the charges and inversely proportional to the distances separating the charge quantities. The distance R is obtained by the vectorial differences of the position coordinates of the two charge quantities (ie components of the equation (ie 4 space. Hence the electrostatic field produced by the charge as force per unit charge. (2) on the test charge q is given R= r ). The other ) is a constant associated with the geometry of</p> <p>In equation (2) above, the limit taken as q tends to zero implies that the test charge q is so small to the extent that it has no effect on the field produced by the charge under study ( ) Substituting equ (1) into equ (2)</p> <p>(3) Here equation (3) is similar to equation (1) above except that the electrostatic field is produced by the charge under study due to the earlier assumption that the quantity of the test charge tends to zero and hence the electric field produced by the test charge is negligible. Suppose that the magnitude of the discrete electric charges are so minute and the number of charges so massive, then the concept of electric charge density introduced. Hence, will be</p> <p>..........................(4) The above equation(4) is exactly the same as equation(3) but while equation (3) considers a point charge, equation (4) considers charge density difficult to study point charges). For the purpose of analysis and study of the property of the electrostatic field E, the Curl and divergence of the field is taken as below. Prior to this analysis, lets look at what divergence ( and Curl ( means. within a volume (It is worthy of note here that in reality, equation (4) is what is obtainable as it is</p> <p>First of all, divergence and curl are both operators used to study vector and scalar fields. Now, the divergence of a vector is the outflow of flux( a flux is nothing other than lines of electric and magnetic fields) from a small closed surface per unit volume as the volume shrinks to zero. Divergence ordinarily tells us how much flux is leaving a small area on a per unit volume basis without any direction. Similarly, the Curl of a vector is the measure of the strength of rotation or vorticity of the vector field around the corresponding coordinate direction. In order to obtain the characteristics of a vector or scalar field, divergence and curl operators are applied to such quantities. Recall, from vector field identities, the curl is equivalent to zero[2] i.e . Applying this in the electrostatic field E. of any well behaved scalar field</p> <p>(5) The physical interpretation of equ (5) is that the electrostatic field is irrotational. This is so because the electrostatic field is a scalar quantity(ie it has magnitude without direction). In all of the equations used here, the subscripts indicate the position coordinate of the vector quantity (for instance, implies the charge</p> <p>density at the position coordinate of (</p> <p>). Similarly the integral sign(</p> <p>in any</p> <p>of the equations implies the summation of the total charge density within the enclosed volume. Similarly, taking the divergence of the electrostatic field E, we have.</p> <p>(6) The above equ(6) is a differential form of Gausss law. That is to say that equation (6) conforms to the earlier statement of Gauss that the total amount of fields radiating from a closed surface is equal to the total number of charges within the same closed volume.</p> <p>1.2</p> <p>ANALYSIS AND DERIVATIONS OF MATHEMATICAL MODELS FOR ELECTRIC FIELD OF DIFFERENT CHARGE DISTRIBUTIONS AND APPLICATION OF GAUSSS LAW</p> <p>FIELD OF N-POINT CHARGES Since coulomb forces are linear, the electric field intensity due to two point charges Q1 and Q2 is the sum of the forces on Q1 caused by Q1 and Q2 acting alone. This also implies that electric fields produced by different point charges are also linear. Mathematically this can be expressed thus:</p> <p>E= OR</p> <p>a</p> <p>+</p> <p>a</p> <p>+ - - - +</p> <p>a</p> <p>FIELD DUE TO A CONTINOUS VOLUME CHARGE DISTRIBUTION The small amount of charges = in a small volume is given by relation . (8)</p> <p>Hence the total charge within some finite volume is given thus: Q = E=</p> <p>E=</p> <p>FIELD OF A LINE CHARGE ( ) In the determination of electric field E at any point resulting from a uniform line charge density , we use symmetry to determine two factors: (a) With which coordinate does the field not vary and (b) which components of the field are not present. The line field varies only with r coordinate. No element of charge produces a Hence = component . Each element does produce an Er and EZ component, but the</p> <p>contributions of charge due to + EZ and - EZ which are equal distances above and below the point at which the field is being determined will cancel out leaving only the Er component, which varies only with r. Hence, OR</p> <p>= .(11) Equation (11) looks at the differential electric field within the length of the sheet, when integrated within the entire length of the wire, equation (12) is obtained. Replacing charge. by and summing the contribution from every element of</p> <p>(12)</p> <p>FIELD OF A SHEET OF CHARGE (SURFACE CHARGE DENSITY</p> <p>)</p> <p>Another basic charge configuration is the infinite sheet of charge having a uniform density of such a charge distribution may be used to approximate that found on the conductors of a strip transmission here that static charges reside on the conductor surfaces and not in their interiors The line charge density or charge per unit length is Where This is the position vector of the charge density(that is the vectorial distance) ..... (13)</p> <p>=</p> <p>-</p> <p>Hence,</p> <p>..(14)</p> <p>The first line of the above equation looks at the differential field dE in the x direction while the second line takes the integral of differential field so as to obtain the total field of the charge distribution within the sheet.</p> <p>1.3 GAUSSS LAW This states that the electric Flux passing through any closed surface is equal to the total charge enclosed by that surface. The total flux ( ) passing through the closed surface is obtained by adding the differential contributions crossing each surface element s That is .. (15)</p> <p>Expressing Gausss statement mathematically, we have (16) Equation (16) implies that total the amount of electric or magnetic flux emanating from an enclosed surface filled with charges is equal to the total number of charges( ) within that volume. This is true because each charge quantity radiates its own field so the total fields from a volume is the same as the sum of the fields from each of the charges within the volume. Where = Electric Flux Density One the applications of Gausss Law that it is used to convert surface integral to volume integral.</p> <p>1.4 APPLICATION OF GAUSS LAW TO SOME SYMETRICAL CHARGE</p> <p>DISTRIBUTIONS</p> <p>The objective here is to use Gauss law to determine (ie the electric flux density) if the charge distribution is known. To be able to do this a closed surface which satisfies two conditions has to be chosen. The conditions are (a) where is everywhere either normal or tangential to the closed surface so that is not zero, becomes or zero respectively. (b) on that portion of the closed surface for which = constant. This allows the replacement of the dot product with the products of the scalars and and then to bring outside the integral sign(ie the above two conditions if met, makes the integration of the resultant integral possible). The remaining integral then becomes over that portion of the closed surface which crosses normally and this is simply the area of this section of that surface. The Gaussian surface for an infinite line charge is a right circular cylinder of length L and radius . Also worthy of mention is the fact that Gauss law cannot be used to obtain solution for electric flux density D if symmetry does not exist. Using Gaussian law, the electric flux density for spherical surface is given thus .(17)</p> <p>Recall Hence, , ie Electric flux density is charge per unit area. , This is the electric field relation for spherical surface charge distribution. The electric flux density ( ) of a uniform line charge for a circular cylinder can be obtained thus</p> <p>, This equation takes a look at the flux density round the entire cylinder, based on the divergence theorem which states that the integral of the normal component of any vector field over a closed surface is equal to the integral of the divergence of this vector field throughout the volume enclosed by the closed surface. From the foregoing, the circular cylinder has the normal component of the electric flux density by the sides. Hence the field at any other part of the cylinder amounts to zero as indicated in the above integral equation.</p> <p>(19) In terms of the charge density Hence, the total charge enclosed is</p> <p>1.5 APPLICATION OF GAUSSS LAW TO A DIFFRENTIAL VOLUME ELEMENT The aim here is to apply Gausss law to obtain electric flux density ( ) for problems that do not possess any symmetry at all. Without symmetry, a simple Gaussian surface cannot be chosen such that the normal component of is constant or zero everywhere on the surface and without such a surface the integral cannot be evaluated. The approach to this problem is to choose such a very small closed surface that is almost constant over the surface and the small change in may be adequately represented using the first two terms of the Taylors series expansions of . The result approaches the correct value as the volume enclosed by the Gaussian surface decreases and approaches zero.</p> <p>Using this approach will not give the exact value of</p> <p>as obtained in the</p> <p>symmetrical case above, rather it will give a useful information about the way varies in the</p> <p>Fig1.1</p> <p>region of our small surface Consider any point P in a given volume as shown below</p> <p>Hence,</p> <p>(20) ..(21)</p> <p>Equations (20) and (21) implies that the charge enclosed in the volume is approximately equal to. Here the approximate result is obtained due to lack of symmetry by the chosen surface thereby making the integral equation unintegrable.</p> <p>1.6 MAGNETOSTATICS This is the second part of electrodynamics which deals with stationary electric currents, that is, electric charges moving with constant speeds, and the interactions between these currents. While coulomb looks at the force produced when a charge imparts on another charge q called the test charge) Ampere shows that electric current interact in much the same way. Let F denotes such a force acting on a small loop C carrying a current J located at r, due to the presence of a small loop C carrying a current located at According to Amperes law, this force in vacuum is given by</p> <p>Hence</p> <p>(22) dL j rr</p> <p>O Fig 1.2 The diagram above illustrates Amperes law.</p> <p>Hence, the differential element dB (r) of the static magnetic field set up, at the field point r by a small line element d of stationary current at the source point ri is shown thus[2]: (23 Taking the integral of equation (23) above:</p> <p>=</p> <p>(24)</p> <p>In order to assess the properties of B, its divergence and curl are determined thus: Taking the divergence of both rides of equation (24)</p> <p>(25) Similarly, taking the curl of equation (24)</p> <p>(26) Thus the uncoupled equations of electrodynamics for E &amp; B fields are as follows. 1.</p> <p>2.</p> <p>3.</p> <p>4. 1.7 UNIFICATION OF ELECTRODYNAMIC THEORY</p> <p>Here, the objective is to derive expressions that will couple the E and B fields. However, the unification of the theories of electrostatics and magneto statics is based on these two empirically established facts: i. Electric charge is a conserved quantity and electric current is a transport of electric charge. This is expressed in the equation of continuity and as a consequence, in max wells displacement current[2]. A charge in the magnetic flux through a loop will induce an electromotive force (emf) in the loop. This is the celebrated Faradays Law of induction[2].</p> <p>ii.</p> <p>Hence, the unified theories of electrodynamics is expressed by the four Maxwells equations given thus: 1) (27)</p> <p>2) 3) 4) 1.8</p> <p>(28) (29) (30)</p> <p>PHYSICAL INTERPRETATION OF MAXWELLS EQUATION 1. Maxwells first equation states that the electric flux per unit volume leaving a vanishingly small volume unit is exactly equal to the volume charge density within that same volume (ie of Gauss Law. = ) this is differential form</p> <p>2. The second equation of Maxwell states that whenever there is a rate of charge of magnetic flux (B) an emf is induced. This statement stems from faradays law (ie )</p> <p>3. The third equation states that the total magnetic flux density within an enclosed surface is equal to zero (ie )</p> <p>4. Maxwells fourth equation implies that the magneto motive force around a closed path (loop) is equal to the total current enclosed by the loop plus the displacement current (ie. ). Thi...</p>