electromagnetiic theory : electrical & communication engineering, the gate academy

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Syllabus EMT

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Syllabus for Electromagnetic Theory

Elements of vector calculus: divergence and curl; Gauss and Stoke’s theorems, Maxwell’s

equations: differential and integral forms. Wave equation, Poynting vector. Plane waves:

propagation through various media; reflection and refraction; phase and group velocity; skin

depth. Transmission lines: characteristic impedance; impedance transformation; Smith chart;

impedance matching; S parameters, pulse excitation. Waveguides: modes in rectangular

waveguides; boundary conditions; cut-off frequencies; dispersion relations. Basics of

propagation in dielectric waveguide and optical fibers. Basics of Antennas: Dipole antennas;

radiation pattern; antenna gain.

Analysis of GATE Papers

(Electromagnetic Theory)

Year Percentage of marks Overall Percentage

2013 4.00

2.19%

2012 2.00

2011 2.00

2010 0.00

2009 0.00

2008 4.00

2007 6.00

2006 0.67

2005 2.00

2004 1.34

2003 6.00

Contents EMT

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page I

CC OO NN TT EE NN TT SS

Chapter Page No

#1. Electromagnetic Field 1 – 49 Introduction to Vector Calculus 1 – 7

Material and Physical Constants 7 – 8

Electromagnetic (EM Field) 8 – 18

Divergence of Current Density and Relaxation 18 – 22

The Magnetic Vector Potential 22 – 27

Faraday Law 27 – 29

Maxwell’s Equation’s 29 – 36

Assignment 1 37 – 39

Assignment 2 40 – 42

Answer keys 43

Exlanations 43 – 49

Module Test 50 – 59 Test Questions 50 – 55

Answer Keys 56

Explanations 56 – 59

Reference Books 60

Chapter 1 EMT

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 1

CHAPTER 1

Electromagnetic Field

Introduction to vector calculus

Cartesian coordinates (x, y, z), x , y , z Cylindrical coordinates ( , , z), , , z Spherical coordinates (r, , ) , r , , Vector calculus formula

Table 1.1 S. No Cartesian coordinates Cylindrical

coordinates Spherical coordinates

(a) Differential displacement dl = dx + dy + dz

dl = d + d

+dz

dl = dr + rd + r sin d

(b) Differential area ds = dydz = dxdz

= dxdy

ds = d dz

= d dz = d d

ds = r sin d d = r sin dr d = r dr d

(c) Differential volume dv = dxdydz

dv = d d dz dv = r sin d d dr

Operators

1) V – gradient , of a Scalar V

2) .V – divergence , of a vector V

3) V – curl , of a vector V 4) V – laplacian , of a scalar V

DEL Operator

=

(Cartesian)

=

(Cylindrical)

=

(Spherical)

Gradient of a Scalar field V is a vector that represents both the magnitude and the direction of maximum space rate of increase of V.

V =

=

=

Chapter 1 EMT


The following are the fundamental properties of the gradient of a scalar field V:

1. The m gnitude of V equ ls the m ximum r te of ch nge in V per unit dist nce. 2. V points in the direction of the maximum rate of change in V. 3. V t ny point is perpendicular to the constant V surface that passes through that point. 4. If A = V, V is s id to be the sc l r potenti l of A. 5. The projection of V in the direction of unit vector |a| is V. |a| and is called the directional

derivative of V along |a|. This is the rate of change of V in direction of |a|.

Example: Find the gradient of the following scalar fields:

(a) V = e sin 2x cosh y (b) U = z cos (c) W = r sin cos

Solution

(a) V =

= e cos x cosh y e sin x sinh y e sin x cosh y

(b) U =

= z cos z sin cos

(c) W =

= sin cos sin cos sin

Divergence of vector

A at a given point P is the outward flux per unit volume as the volume shrinks about P.

Hence,

divA = . A = lim ∮ .

(1)

Where, V is the volume enclosed by the closed surf ce S in which P is loc ted. Physic lly, we

may regard the divergence of the vector field A⃗⃗ at a given point as a measure of how much the field diverges or emanates from that point.

.A =

=

( A )

=

(r A )

(A sin )

From equation (1), ∮ A ds

= ∫ . A dv

Chapter 1 EMT


This is called divergence theorem which states that the total outward flux of the vector field A through a closed surface S is same as the volume integral of the divergence of A. Example

Determine the divergence of these vector field:

(a) P = x yz xz (b) Q = sin z z cos

(c) T =

cos r sin cos cos

Solution

(a) P =

P

P

P

=

x(x yz)

y( )

z(xz)

= xyz x

(b) Q =

( Q )

Q

Q

=

( sin )

( z)

z (z cos )

= sin cos

(c) T =

(r T )

(T sin )

(T )

=

r

r(cos )

r sin

(r sin cos )

r sin

(cos )

=

r sin r sin cos cos

= cos cos

Curl of a vector field provides the maximum value of the circulation of the field per unit area and indicates the direction along which this maximum value occurs.

That is,

curl A = A = lim (∮ .

)

------------- (2)

A = |

A A A

|

=

|

A A A

|

Chapter 1 EMT


=

|

r r sin

A rA r sin A

|

From equation (2) we may expect that ∮ . = ∫ (

).

This is called stoke’s theorem, which states that the circulation of a vector field A around a (closed) path L is equal to the surface integral of the curl of A over the open surface S bounded by L. Example Determine the curl of each of the vector fields of previous Example. Solution

(a) = (

) (

) (

)

= ( ) ( ) ( )

= ( )

(b) = *

+ *

+

*

( )

+

= (

) ( )

( )

=

( ) ( )

(c) =

*

( )

+

*

( )+

*

( )

+

=

[

( )

( )]

[

( )

( )]

[

( )

( )

]

=

( )

( )

(

)

= (

)

(

)

(a) Laplacian of a scalar field V, is the divergence of the gradient of V and is written as .

=

=

(

)

=

(

)

(

)

If = 0, V is said to be harmonic in the region. A vector field is solenoid if .A = ; it is irrot tion l or conserv tive if A =

. ( ) = ( ) =

Chapter 1 EMT


(b) Laplacian of vector A̅

A⃗⃗ = is lw ys vector qu ntity

A⃗⃗ = ( A ) ̂x ( A ) ̂y ( A ) ̂z

A Sc l r qu ntity A Sc l r qu ntity

A Sc l r qu ntity

V =

........Poission’s Eqn

V = ........Laplace Eqn

E = ⃗⃗

E

....... wave Eqn

Example

The potential (scalar) distribution is given as

V = y x if E0 : permittivity of free space what is the change density p at the point (2,0)?

Solution

Poission’s Eqn V =

(

) ( y x ) =

x x x x x y =

At pt( , ) x x x =

=

Example

Find the Laplacian of the following scalar fields,

(a) V = e sin 2x cosh y (b) U = z cos (c) W = r sin cos

Solution

The Laplacian in the Cartesian system can be found by taking the first derivative and later the second derivative.

(a) V =

=

x( e cos x coshy)

y(e sin x sinh y)

z( e sin x cosh y)

= e sin x cosh y e sin x coshy e sin x cosh y = e sin x cosh y

(b) U =

(

)

Chapter 1 EMT


=

( z cos )

z cos

= z cos z cos =

(c) W =

(r

)

(sin

)

=

r

r( r sin cos )

r sin

( r sin sin cos )

r sin cos

r sin

= sin cos

r

r cos sin cos

r sin

r sin cos cos

r sin

cos

r

= cos

r ( sin cos cos )

= cos

r( cos )

Stoke’s theorem

Statement:- closed line integral of any vector A⃗⃗ integrated over any closed curve C is always

equal to the surface integral of curl of vector A⃗⃗ integr ted over the surf ce re ‘s’ which is enclosed by the closed curve ‘c’

∮ A⃗⃗ . d ⃗ = ∫ ∫( x A⃗⃗ ) dS⃗

The theorem is valid irrespective of

(i) Shape of closed curve ‘C’ (ii) Type of vector ‘A’ (iii) Type of co-ordinate system. Divergence theorem

∯ A⃗⃗

dS⃗ = ∭ V⃗⃗ . A⃗⃗ dv

S

V

S

C

Chapter 1 EMT


Statement

Closed surface integral of any vector A⃗⃗ integrated over any closed surface area. S is always equal

to the volume integral of the divergence of vector A⃗⃗ integrated over the volume V which is enclosed by the closed surf ce re ‘S’ the theorem holds good, irrespective (i) Shape of closed surface (ii) Type of coordinate system

(iii) Type of vector A⃗⃗

Material & Physical constants

(a) Material constants

Table 1.2 Material Conductivity ( ) S/m Relative Permittivity ( r) Air 0 1.0006 Aluminum 3.186 107 1.0 Bakelite 10-14 5 Brass 2.564 107 1 Carbon 3 104 Copper 5.8 107 1 Glass 10-13 6 Graphite 105 Mica 10-15 6 Paper 3 Paraffin 10-15 2.1 Plexiglas 3.4 Polystyrene 10-16 2.7 PVC 2.7 Porcelain 5 Quartz 10-17 5 Rubber 10-13 5 Rutile 100 Soil(clay) (sandy)

5 10-3

2 10-3 14 10

Urban ground 2 10-4 4 Vaseline 2.2 Terflon 10-15 2.1 Water (distilled) (fresh) (sea)

10-4

10-2 to 10-3 4 to 5

80 80 80

Wood 2 Transformer oil 2 to 3 Ebonite 2.6 Epoxy 4

electromagnetiic theory : electrical & communication engineering, the gate academy

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