ELECTROMAGNETIC THEORY
for Electrical Engineering By
www.thegateacademy.com
Syllabus
EMT
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
2013
4.00
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
Overall Percentage
2.19%
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Contents
EMT
CONTENTS
#1.
Chapter Electromagnetic Field
Introduction to Vector Calculus Material and Physical Constants Electromagnetic (EM Field) Divergence of Current Density and Relaxation The Magnetic Vector Potential Faraday Law Maxwell’s Equation’s Assignment 1 Assignment 2 Answer keys Exlanations
Module Test
Test Questions Answer Keys Explanations
Reference Books
Page No 1 – 49 1–7 7–8 8 – 18 18 – 22 22 – 27 27 – 29 29 – 36 37 – 39 40 – 42 43 43 – 49
50 – 59 50 – 55 56 56 – 59
60
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Chapter 1
EMT
CHAPTER 1 Electromagnetic Field
Introduction to vector calculus Cartesian coordinates (x, y, z), Cylindrical coordinates ( , , z), Spherical coordinates (r, , ) ,
x r
,
, ,
y ,
,
,
z
z
Vector calculus formula Table 1.1 S. No (a) (b)
(c)
Cartesian coordinates
Cylindrical coordinates + d Differential displacement dl = d dl = dx + dy + dz +dz ds = d dz Differential area ds = dydz = d dz = dxdz = d d = dxdy Differential volume dv = d d dz dv = dxdydz
Spherical coordinates dl = dr + rd + r sin d ds = r sin d d = r sin dr d = r dr d dv = r sin d d dr
Operators 1) 2) 3) 4)
V – gradient , of a Scalar V .V – divergence , of a vector V V – curl , of a vector V 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= = = THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 1
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 ⃗ at a given point as a measure of how much the may regard the divergence of the vector field A field diverges or emanates from that point. .A = =
( A )
=
(r A )
(A sin )
From equation (1), ∮ A ds = ∫ . A dv THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 2
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 (b) Q = sin (c) T =
xz z
cos
z cos
r sin cos
cos
Solution (a)
P=
P
P
(b)
(x yz) x = xyz x Q= ( Q )
P
=
(
= (c)
y
( )
z
Q
sin )
r
r
z
(z cos )
(T sin )
(cos )
=
Q ( z)
= sin cos (r T ) T= =
(xz)
(r sin
r sin
(T ) cos )
r sin
(cos )
r sin cos cos
r sin = 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=| A
= | A
∮
A = lim
(
.
)
------------- (2)
| A
A
| A
A
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Chapter 1
r =
EMT
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=
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Chapter 1
EMT
̅ (b) Laplacian of vector A ⃗A = is lw ys vector qu ntity ⃗A = ( 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= )(
( x x x
x )=
x x y =
)
At pt( ,
y
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= (e sin x sinh y) ( e cos x cosh y) ( e x y z e sin x cosh y e sin x cosh y e sin x cosh y e sin x cosh y
= = = (b)
U=
(
sin x cosh y)
)
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Chapter 1
(
=
= z cos = (c)
W= = = = =
r
)
z cos
z cos
z cos
(r
)
(
r sin
r sin
EMT
(sin
)
cos )
cos
r cos ( sin r cos ( cos r
r sin r cos sin cos r sin cos
cos
(
r sin
sin cos )
r sin cos cos r sin
r sin cos r sin cos r
)
)
Stoke’s theorem ⃗ integrated over any closed curve C is always Statement:- closed line integral of any vector A ⃗ integr ted over the surf ce re ‘s’ which is equal to the surface integral of curl of vector A enclosed by the closed curve ‘c’ S
C
⃗ . d⃗ = ∫ ∫( x A ⃗ ) dS⃗ ∮A
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 S
V ∯ ⃗A dS⃗ = ∭ ⃗V. ⃗Adv
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Chapter 1
EMT
Statement Closed surface integral of any vector ⃗A integrated over any closed surface area. S is always equal to the volume integral of the divergence of vector ⃗A 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 ⃗A
Material & Physical constants (a) Material constants Table 1.2 Material Air Aluminum Bakelite Brass Carbon Copper Glass Graphite Mica Paper Paraffin Plexiglas Polystyrene PVC Porcelain Quartz Rubber Rutile Soil(clay) (sandy) Urban ground Vaseline Terflon Water (distilled) (fresh) (sea) Wood Transformer oil Ebonite Epoxy
Conductivity ( ) S/m 0 3.186 107 10-14 2.564 107 3 104 5.8 107 10-13 105 10-15 10-15 10-16 10-17 10-13 5 10-3 2 10-3 2 10-4 10-15 10-4 10-2 to 10-3 4 to 5
Relative Permittivity ( r) 1.0006 1.0 5 1 1 6 6 3 2.1 3.4 2.7 2.7 5 5 5 100 14 10 4 2.2 2.1 80 80 80 2 2 to 3 2.6 4
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