## Vector Algebra

• Chapter 1: Vector Algebra . %1.1Introduction . –Definition: Electromagnetics (EM): The study of the interaction between electric charges at rest and in motion. It entails the analysis, synthesis, physical interpretation and application of electric and magnetic fields. In other words, electromagnetics (EM) is a branch of physics or electrical engineering in which electric and magnetic phenomena are studied. –Examples of applications: 1.Microwaves 2.Antennas 3.Electric machines 4.Satellite communications 5.Bioelectromagnetics 6.Plasmas 7.Nuclear research 8.Fiber optics 9.Electromagnetics interference and compatibility 10.Electromechanical energy conversion 11.Radar meteorology 12.Remote sensing Shortwaves . –In physical medicine: Electromagnetics power Microwaves . –Examples of devices: 1.Transformers 2.Electric relays 3.Radio/TV 4.Telephone 5.Electric motors 6.Transmission lines 7.Waveguides 8.Antennas 9.Optical fibers 10.Radars 11.Lasers . . . –.Summary of the subject of this book: Where, : The vector differential operator : The electric flux density : The magnetic flux density : The electric field intensity : The magnetic field intensity : The volume charge density : The current density . Maxwell’s equations:     . . . %1.3Scalar and Vectors . –Definitions: Scalar: A quantity that has only magnitude. Examples: Time, mass, distance, temperature, entropy, electric potential, population. . Vector: A quantity that has both magnitude and direction. Examples: velocity, force, displacement, electric field intensity. *NOTE: Another class of physical quantities is called Tensors, of which scalars and vector are special cases.   Field: A function that specifies a particular quantity everywhere in a region. Examples:  - Scalar fields: Temperature distribution, sound intensity, electric potential, refractive index of a stratified medium. – Vector Fields: Gravitational force, velocity. . . . %1.4Unit Vectors . –Notes about notations: A vector as said above, has both magnitude and direction. The magnitude is written as or The vector it self as .Now, a unit vector along is defined as a vector whose magnitude is unity and its direction is along . That is,     . and thus,   . . . .  The vector in Cartesian (also called rectangular) coordinates may be represented as:      . .  And the magnitude of is found as follows:    .  and   . . . . . %1.5Victor Addition and Subtraction . . . . . . . –Where k and are scalars . . . . . . . . . %1.6Victor Addition and Subtraction . –Definitions: .Position vector (radius vector): The position vector of a point P is defined as the directed distance from the origin O to to P. That is,     . Distance vector (separation vector): The...

## Dimplementationofmaxwell’s Equationsin Matlabtostudythe Effect Of Absorption Using Pml

1-D Implementation of Maxwell’s Equations in MATLAB to Study the Effect of Absorption Using PML . Pranav K Shrivastava,Vikas Rathi, Hemant S Pokhariya Graphic Era Unversity ,Dehradun e-mail:{pranav.shantikunj,vikas.rth}@gmil.com . . Abstract: The Finite Difference Time Domain method   (FDTD)     uses     centre-difference representations   of   the   continuous   partial The 3D source free( J =0) Mxwell’s curl equations  a homogeneous medium are: . ∂Ez − ∂Ey = −µ ∂Hx ∂y differential equations to create iterative numerical  dH  ∂Ex ∂z ∂Ez ∂t ∂Hy models of wave propagation. First we study the ∇×E = −µ → ∂z   − ∂x   = −µ ∂t propagation behavior of the wave in single dt  ∂Ey ∂Ex ∂Hz dimension without PML and in second part we study the absorption using PML for the same wave  ∂x   − ∂y = −µ ∂t using MATLAB environment. . I. INTRODUCTION . Finite-difference  time-domain  (FDTD)  is  a  popular computational electrodynamics modeling technique. Since it is a . ∇ × H = ε  ∂Hz d E      ∂Hx dt    ∂Hy →   ∂y ∂z  ∂x ∂Hy −   ∂z    = ε ∂Hz = −    ∂t  ε ∂Hx −   ∂y    = ε ∂Ex ∂t ∂Ey ∂t ∂Ez ∂t time-domain method, solutions can cover a wide frequency range with a single simulation run. ∂Ex   = −µ ∂Hy ∂z  ∂t ∂Hy   = −ε ∂Ex The FDTD method belongs in the general class of grid-based differential time-domain numerical modeling methods. The time- dependent Maxwell’s equations (in partial differential form) are discretized using central-difference approximations to the space and time partial   derivatives. The resulting finite-difference equations are solved in either software or hardware in a leapfrog manner: the electric field vector components in a volume of space are solved at a given instant in time; then the magnetic field vector components in the same spatial volume are solved at the next instant in time; and the process is repeated over and over again until the desired transient or steady-state electromagnetic field behavior is fully evolved. . When Maxwell’s differential equations are examined, it can be seen that the change in the E-field in time (the time derivative) is dependent on the change in the H-field across space (the curl). This results in the basic FDTD time-stepping relation that, at any point in space, the updated value of the E-field in time is dependent on the stored value of the E-field and the numerical curl of the local distribution of the H-field in space. . The H-field is time-stepped...

UNIT III DIMENSIONAL ANALYSIS . . . . UNIT III DIMENSIONAL ANALYSIS . The basic concepts and procedures for dimensional analysis were developed by hydraulic engineers to determine the performances of a prototype (a full-scale structure) from the data obtained by tests on a model ( a reduced-scale structure). Here we present the general method of dimensional analysis and illustrates its application to various problems of fluid machines. Some of the important principles of similarity and use of dimensionless numbers in model analysis are also studied. . . SYSTEM OF DIMENSIONS: Dimensions refer to the qualitative characteristics for physical quantities, while units are standards of comparison for quantitative measure of dimensions. The most common systems of dimensioning a physical quantity and the Mass-length-time and the Force-length-time systems referred to as the MLT and FLT systems of units. There is no direct relationship between the quantities length L, mass M and time T. These independent quantities are called fundamental quantities. In compressible fluids, one more dimension namely temperature θ is also taken as the fundamental dimension. All other quantities such as pressure, velocity and energy etc. are expressed in terms of these fundamental quantities and are called derived or secondary quantities. For example F=MLT−2:M=FT2L−1 . . Physical quantity Symbol Dimensions M-L-T System F-L-T System *Fundamental quantities . . . Mass M M FL-1T2 Length L L L Time T T T Force F MLT-2 F *Geometric quantities . . . Area A L2 L2 Volume V L3 L3 *Kinematic quantities . . . Linear velocity u,V,U LT-1 LT-1 Angular velocity ω T-1 T-1 Acceleration a LT-2 LT-2 Discharge Q,q L3 T-1 L3 T-1 Gravity g LT-2 LT-2 Kinematic viscosity ν L2 T-1 L2 T-1 *Dynamic quantities . . . Density ρ ML-3 FL-4T2 Specific Weight w ML-2T-2 FL-3 Surface tension σ MT-2 FL-1 Pressure intensity p ML-1T-2 FL-2 Modulus of elasticity E, K ML-1T-2 FL-2 Dynamic viscosity μ ML-1T-1 FL-2T Resisting force F, R MLT-2 F Thrust T MLT-2 F Torque T ML2T-2 FL Work W ML2T-2 FL Energy E ML2T-2 FL Power P ML2T-3 FLT-1 . . . . DIMENSIONAL HOMOGENEITY AND ITS APPLICATIONS: The fundamental theory of dimensional analysis is based on the following axiom: “Equations describing a physical phenomenon must dimensionally homogeneous and the units therein must be consistent”. The concept of dimensional homogeneity can be elaborated by considering the time period of oscillation T of a simple...

## Binomial and Normal distributions used in business forecastingMade By:Abhay SinghRoll No.—50202BBS I ABusiness Statistics and ApplicationsTerm Paper

statistics term paper Figure 1 1 What are theoretical distributions? 3 Binomial Distribution 3 Cumulative Distribution Function 4 Why is it important? 4 (p + q)2 = p2 + 2pq +q2, or more simply, pp + 2pq + qq 4 Quick facts 4 Uses in Business 5 1. Quality Control 5 2. Public Opinion Survey 6 3. Medical Research 6 4. Insurance Sector 7 Normal Distribution 8 Uses 10 1. Modern Portfolio Theory 10 2. Human resource Management 11 Forecasting 11 1. Using the Normal Distribution to Determine the Lower 10% Limit of Delivery Times 13 2. Finding the probability of a certain type of package passing down a conveyor belt if the probability of that type of package passing by is known. 14...