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Electro Magnetic Field Theory (EE-402)Syllabus: Chapter1: Introduction, Vector analysis, Co-ordinate systems and transformation, Cartesian coordinates, Circular cylindrical coordinates, spherical coordinates & their transformation. Differential length, area and volume in different coordinate systems. 3 Chapter2: Introduction to vector calculus: Del operator, Gradient of a scalar, Divergence of a vector & Divergence theorem, Curl of a vector & Strokes theorem. Laplacian of a scalar. Classification of vector fields. Helmholtzs theorem. 3 Chapter3: Electro static field: Coulombs law, field intensity, Gausss law- Maxwells equation, Electric potential and potential gradient, Relationship between E and V-Maxells equation An electric Dipole & flux lines, Energy density in electrostatic fields. Boundary conditions: Dielectric-dielectric, Conductor-dielectric, Conductor-free space, Poissons and Laplaces equations, General procedure for solving Poissons and Laplaces equation. 8 Chapter4: Magneto static fields: Biot-Savart Law, Amperes Circuit law-Maxwells equation, Magnetic Flux density-Maxwells equation, Maxwells Equation for static fields, Magnetic static and vector potential, forces due Magnetic fields, Magnetic torque and moments, Magnetization in material, Magnetic boundary condition, inductor and inductances, Magnetic energy, Force on magnetic materials. 10 Chapter5: Electromagnetic field: Faradays law, Transformer and motional EMF, Displacement current, Maxwells equations, Time varying potentials, Time harmonic fields. 4 Chapter6: Electromagnetic wave propagation: Wave propagation in lossy dielectrics, plane waves in lossless dielectric, plane wave in free space, plane wave in good conductor, skin effect, skin depth, power and the Poynting vector, reflection of a plane wave at normal incidence, reflection of a plane wave at oblique incidence, Polarisation. 7 Chapter7: Transmission lines, Transmission line parameters, Transmission line equation. Books: 1. Electromagnetic field theory, Gangadhar 2. Theory & Problems in Electromagnetic, 2/e, Edminister,TMH 3. Engineering Electromanetics, 7/e, Hyat, TMH 3

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Chapter1Introduction: Electromagnetic field theory is a discipline concerned with the study of charges at rest and in motion. Electromagnetic principles are fundamental to the study of electrical engineering. Electromagnetic theory is also indispensable to the understanding, analysis and design of various electrical, electromechanical and electronic systems. Some of the branches of study where electromagnetic principles find application are: RF communication, Microwave Engineering, Antennas, Electrical Machines, Satellite Communication, Atomic and nuclear research, Radar Technology, Remote sensing, EMI EMC, Quantum Electronics, VLSI Electromagnetic theory is a prerequisite for a wide spectrum of studies in the field of Electrical Sciences. Electromagnetic theory can be thought of as generalization of circuit theory. There are certain situations that can be handled exclusively in terms of field theory. In electromagnetic theory, the quantities involved can be categorized as source quantities and field quantities. Source of electromagnetic field is electric charges: either at rest or in motion. However an electromagnetic field may cause a redistribution of charges that in turn change the field and hence the separation of cause and effect is not always visible. Electric charge is a fundamental property of matter. Charge exist only in positive or negative integral multiple of electronic charge, -e, e= 1.60 10-19 coulombs. [It may be noted here that in 1962, Murray Gell-Mann hypothesized Quarks as the basic building 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. Vector Analysis 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 a Vector Analysis: The quantities that we deal in electromagnetic theory may be either scalar or vectors [There are other classes 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. Representation of a Vector: A vector can be written as , where, is the magnitude and and

is the unit vector which has unit magnitude and same direction as that of . Two vector are added together to give another vector . We have

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Scaling of a vector is defined as , where Some important laws of vector algebra are:

is scaled version of vector

and

is a scalar.

Commutative Law..........................................(1.3) Associative Law.............................................(1.4) Distributive Law ............................................(1.5) Position vector & Distance vector: The position vector origin (O) to P, i.e., = . of a point P is the directed distance from the

Fig 1.3: Distance Vector If = OP and = OQ are the position vectors of the points P and Q then the distance vector

Representation of vector in Cartesian co-ordinate system: is vector directed from the origin O to the point P(x,y,z). The unit vector along the positive direction of X, Y and Z-axis are i, j and k respectively. Then the vector

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Z

z P ( x , y ,z ) R

O

y

Y

x

X

Addition of vectors: Addition of vectors can be done either by using the laws (Triangular and Parallelogram law) or by representing the vectors in co-ordinate system form. ExampleThen and = i(x1+x2) +j (y1+y2) +k (z1+z2)

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) gives a scalar. The dot product between two vectors is defined as = |A||B|cos, where is the angle between the two vectors. The dot product is commutative i.e., and distributive i.e., = (ix1+jy1+kz1).(ix2+jy2+kz2)= x1.x2+y1.y2+z1.z2 i.i=1, j.j=1, k.k=1and i.j=j.k=k.i=0

Vector product (or cross product) unit vector perpendicular to magnitude is given by

gives a vector. Vector product

, where

is

and . is a vector perpendicular to the plane containing and direction is given by right hand rule.

and , the

=(ix1+jy1+kz1)x(ix2+jy2+kz2)=k(x1y2-y1x2)+j(z1x2-x1z2)+i(y1z2-z1y2)

=

=

ixi=jxj=kxk=0, ixj=k, jxk=i, kxi=j, jxi= -k, kxj= -i, ixk= -j The following relations hold for vector product.Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC 4

=

i.e., cross product is non commutative i.e., cross product is distributive i.e., cross product is non associative

Scalar and vector triple product : Scalar triple product Vector triple product

Co-ordinate Systems 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 nonorthogonal. 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. Further, let , and be the unit vectors in the three coordinate directions(base vectors). In a general right handed orthogonal curvilinear system, the vectors satisfy the following relations:

These equations are not independent and specification of one will automatically imply the other two. Furthermore, the following relations hold

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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A vector can be represented as sum of its orthogonal components, In general u, v and w may not represent length. We multiply u, v and w by conversion factors h1, h2 and h3 respectively to convert differential changes du, dv and dw to corresponding changes in length dl1, dl2, and dl3. Therefore

In the same manner, differential volume dv can be written as normal to is given by

and differential area ds1 and

. In the same manner, differential areas normal to unit vectors

can be defined. Cartesian Co-ordinate System: In Cartesian co-ordinate system, we have, (u, v, w) = (x, y, z). A point P(x0, y0, z0) in Cartesian coordinate system is represented as intersection of three planes x = x0, y = y0 and z = z0. The unit vector satisfies the following relation:

In Cartesian co-ordinate system, a vector product of two vectors and

can be written as

. The dot and cross

can be written as follows:

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Since x, y and z all represent lengths, h1= h2= h3=1. The differential length, area and volume are defined respectively as

Cylindrical Co-ordinate System: For cylindrical coordinate systems we have

a point

is determined as the point of intersection of a cylindrical surface r = r0, half plane containing the z-axis and making an angle with the xz plane and a plane parallel to xy plane located at z=z0 as shown in figure. In cylindrical coordinate system, the unit vectors satisfy the following relations : A vector can be written as ,

The differential length is defined as,

Differential areas are:

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Differential volume,

Following relations are found from the diagram

Fig : Cylindrical Coordinate System and Differential Volume Element

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Spherical Polar Coordinates: For spherical polar coordinate system, we have the intersection of (i) Spherical surface r=r0 (ii) Conical surface , Where is varying between 0 and , (iii) Half plane containing z-axis making angle with the xz plane as shown in figure. varies between 0 and 2 The unit vectors satisfy the following relationships: the . A point is represented as

The orientation of the unit vectors are shown in the figure below.

Fig: Orientation of Unit Vectors A vector in spherical polar co-ordinates is written as: andAuthor- BIKASH PATEL, Assistant Professor, EE Dept, KGEC 9

For spherical polar coordinate system we have h1=1, h2= r and h3=

.

Fig 1.12(b) : Exploded view Fig 1.12(a) : Differential volume in spherical coordinates With reference to the Figure 1.12, the elemental areas are:

and elementary volume is given by

From the diagram it is found the following relations:

Transformation between Cartesian and Cylindrical coordinates: Let us consider is . In doing so we note that and it applies for other components also. to be expressed in Cartesian co-ordinate as

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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From the diagram above it is clear that,

Fig. co-ordinate transformation Therefore we can write

These relations can be put conveniently in the matrix form as:

Where , and z are related to x,y and z as

The inverse relationships are:

Similarly we can transformed a Cartesian co-ordinate into a cylindrical co-ordinate as below:

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Coordinate transformation between Cartesian and spherical co-ordinate system: With reference to the figure 1.13 ,we can write the following equations:

Fig 1.13: Coordinate transformation Given a vector in the spherical polar coordinate system, its component in the Cartesian coordinate system can be found out as follows:

The above equation can be put in a compact form:

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Where

are related to x,y and z as:

and conversely,

Similarly we can transformed a Cartesian co-ordinate into a spherical co-ordinate as below:

Using the variable transformation listed above, the vector components, which are functions of variables of one coordinate system, can be transformed to functions of variables of other coordinate system and a total transformation can be done. Table: Relation between different types of co-ordinate system Curvilinear u v w Cartesian x y z Cylindrical r z Spherical r 13

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

1. Given the vectors Find :a. The vector C = A + B at a point P (0, 2,-3). b. The component of A along B at P

Solution: The vector B is cylindrical coordinates. This vector in Cartesian coordinate can be written as: Where

The point P(0,2,-3) is in the y-z plane for whicha. C = A + B

.

= =b. Component of A along B is

where is the angle between A is and B.

i.e.,

=

2. A vector field is given by

Transform this vector into rectangular co-ordinates and calculate its magnitude at P(1,0,1). Solution: Given, The components of the vector in Cartesian coordinates can be computed as follows:

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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3. The coordinates of a point P in cylindrical co-ordinates is given by

. Find the volume of the sphere that has center at the origin and on which P is a point. If O represents the origin, what angle OP subtends with z-axis? Solution:

The radius of the sphere on which P is a point is given by

Therefore, the volume of the sphere

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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Further

4. A vector field is given by

. Find the value of the Ey component at a point (-2, 3, 1).

Solution:

But we know that

Therefore At the point (-2, 3, 1)

5. An electric field expressed in spherical polar coordinates is given by

. Determine

and

at a point

.

Author- BIKASH PATEL, Assistant Professor, EE Dept, KGEC

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