S J P N Trust's

Hirasugar Institute of Technology, Nidasoshi.Inculcating Values, Promoting Prosperity

Approved by AICTE, Recognized by Govt. of Karnataka and Affiliated to VTU Belagavi

ECE Dept.

EE

III Sem

2018-19

Department of Electronics & Communication Engg.

Course : Engineering Electromagnetics -17EC36. Sem.: 3rd (2018-19)

Course Coordinator:

Prof. S. S. KAMATE

Vector Analysis

Scalars and Vectors

Scalar Fields (temperature)

Vector Fields (gravitational, magnetic)

Vector Algebra

There are three co-ordinate systems

1. Cartesian or rectangular co-ordinate system

2. Cylindrical co-ordinate System

3. Spherical co-ordinate System

The Cartesian Coordinate System : vertices are x,y,z

Vector Components and Unit Vectors

Constant planes

Contd…

Distance vector - dl

Differential surfaces -

dv =

The Dot product

A.B = |A||B| cos qAB

B in the direction of A

You need to normalize a

before the dot product.

The Cross Product

Example

A

2

3

1

B

4

2

5

A B

13

14

16

A B

ax

Ax

Bx

ay

Ay

By

az

Az

Bz

A x B = aN|A||B| sin qAB

Circular Cylindrical Coordinate System

Differential volume

Circular Cylindrical Coordinate

System

x cos

y sin

z z

x2

y2

0

atany

x

z z

A Ax ax Ay ay Az az

A A a A a Az az

A A a A A a Az Az

A Ax ax Ay ay Az az( ) a Ax ax a Ay ay a

A Ax ax Ay ay Az az( ) a Ax ax a Ay ay a

Az Ax ax Ay ay Az az( ) az Az az az Az

az a az 0

Dot

Product

Spherical co-ordinate system

The Spherical Coordinate System

x r sin q cos

y r sin q sin

z r cos q

r x2

y2

z2

r 0

q acosz

x2

y2

z2

0 q 180

atany

x

Sperical Co-ordinate System

The Spherical Coordinate System

x r sin q cos

y r sin q sin

z r cos q

r dr dq

r sin q dr d

r2

sin q dq d

r2

sin q dr dq d

The Experimental Law of Coulomb

F kQ1 Q2

R2

k1

4 00 8.854 10

12

1

36 10

9

F

mF

Q1 Q2

4 0 R2

FQ1 Q2

4 0 R2

a12 a12

r2 r1

r2 r1

Electric Field IntensityFt

Q1 Qt

4 0 R1t 2

a1t

Ft

Qt

Q1

4 0 R1t 2

a1t

EFt

Qt

E r( )Q

4 o r r1 2

r r1

r r1

E r( )Q x x1( ) ax y y1( ) ay z z1( ) az

4 o x x1( )2

y y1( )2

z z1( )2

3

2

Electric Field Intensity

E r( )

1

n

m

Qm

4 0 r rm 2

am

Field of a Line Charge

E

zL

4 0 2

z2

3

2

d

EL

2 0

EL

2 0 a

Field of a Line Charge (neglect symmetry)

E z1y1x1vq

4 0

r r1( )

r r1 3

d

d

d

v1 L dz1

r a z az r1 z1 az

R r r1 a z z1( ) az

R 2

z z1( )2

aR a z z1( ) az

2

z z1( )2

E

z1L dz1 a z z1( ) az

4 0 2

z z1( )2

3

2

d

EL

4 0 a

z1 dz1

2

z z1( )2

3

2

d az

z1z z1( )( )

2

z z1( )2

3

2

d

to to

EL

4 0 a

1

2

z z1( )

2

z z1( )2

az1

2

z z1( )2

EL

4 0 a

2

az 0

L

2 0 a

Field of a Line Charge (neglect symmetry)

Field of a Sheet of Charge

R x2

y2

dEx sdy1

2 0 x2

y2

cos q s

2 0 x dy1

x2

y2

to

Exs

2 0

y1x

x2

y2

ds

2 0 atan

y1

x

Exs

2 0E

s

2 0aN

This is a very interesting result. The field is constant in magnitude and

direction. It is as strong a million miles away from the sheet as it is right of the

surface.

Streamlines and Sketches of Fields

Cross-sectional

view of the line

charge.

Lengths

proportional to the

magnitudes of E

and pointing in the

direction of E

Streamlines and Sketches of Fields

Ey

Ex

dy

dz

E1

a E

x

x2

y2

axy

x2

y2

ay

dy

dx

Ey

Ex

y

x

dy

y

dx

x

ln y( ) ln x( ) C1 ln y( ) ln x( ) ln c( )

y C x

3.1 Electric Flux Density

• Faraday’s Experiment

Electric Flux Density, D

• Units: C/m2

• Magnitude: Number of flux lines (coulombs)

crossing a surface normal to the lines divided

by the surface area.

• Direction: Direction of flux lines (same

direction as E).

• For a point charge:

• For a general charge distribution,

D3.1

Given a 60-uC point charge located at the origin, find

the total electric flux passing through:

(a) That portion of the sphere r = 26 cm bounded by

0 < theta < Pi/2 and 0 < phi < Pi/2

Gauss’s Law

• “The electric flux passing through any

closed surface is equal to the total

charge enclosed by that surface.”

• The integration is performed over a

closed surface, i.e. gaussian surface.

• We can check Gauss’s law with a point

charge example.

Symmetrical Charge Distributions

• Gauss’s law is useful under two conditions.

1. DS is everywhere either normal or tangential to the closed surface, so that DS

.dS becomes either DS

dS or zero, respectively.

1. On that portion of the closed surface for which DS

.dS is not zero, DS = constant.

Gauss’s law simplifies the task of finding D near an infinite line

charge.

Infinite coaxial cable:

Differential Volume Element

• If we take a small enough closed surface,

then D is almost constant over the surface.

Divergence

Divergence is the outflow of flux from a small closed surface area (per unit volume) as volume shrinks to zero.

-Water leaving a bathtub

-Closed surface (water itself) is essentially incompressible

-Net outflow is zero

-Air leaving a punctured tire

-Divergence is positive, as closed surface (tire) exhibits net outflow

Mathematical definition of divergence

div D D x

x

D y

y

D z

z

- Cartesian

div D 0v

SD

v

dlim

Surface integral as the volume element (v) approaches zero

D is the vector flux density

Cylindrical

Spherical

div D 1

D

1

D

Dz

z

div D 1

r2

D r r2

r

1

r sin q

D q sin q q

1

r sin q

D

Divergence in Other Coordinate Systems

4.1 Energy to move a point charge

through a Field

• Force on Q due to an electric field

• Differential work done by an external

source moving Q

• Work required to move a charge a finite

distance

F E QE

dW QE dL

Line Integral

• Work expression without using vectors

EL is the component of E in the dL direction

• Uniform electric field density

W Q

initial

final

LE L

d

W QE LBA

Potential

• Measure potential difference between a

point and something which has zero

potential “ground”VAB VA VB

Potential Field of a Point Charge

• Let V=0 at infinity

• Equipotential surface:

– A surface composed of all points having

the same potential

Potential due to n point

chargesContinue adding charges

V r( )Q1

4 0 r r1

Q2

4 0 r r2 ....

Qn

4 0 r r n

V r( )

1

n

m

Qm

4 0 r r m

Potential as point charges

become infinite

Volume of charge

Line of charge

Surface of charge

V r( ) v prime

v r prime 4 0 r r prime

d

V r( ) Lprime

L r prime 4 0 r r prime

d

V r( ) S prime

S r prime 4 0 r r prime

d

Potential as point charges

become infinite

Volume of charge

Line of charge

Surface of charge

V r( ) v prime

v r prime 4 0 r r prime

d

V r( ) Lprime

L r prime 4 0 r r prime

d

V r( ) S prime

S r prime 4 0 r r prime

d

Gradients in different

coordinate systems

The following equations are found on page 104 and inside the back

cover of the text:

Cartesian

Cylindrical

Spherical

gradVV

xa x

V

ya y

V

za z

gradVV

a

1

V

a

V

za z

gradVV

ra r

1

r

V

q a q

1

r sin q

V

a

Chapter 7 – Poisson’s and Laplace Equations

A useful approach to the calculation of electric potentials

Relates potential to the charge density.

The electric field is related to the charge density by the divergence

relationship

The electric field is related to the electric potential by a gradient

relationship

Therefore the potential is related to the charge density by Poisson's equation

In a charge-free region of space, this becomes Laplace's equation

57

Magnetic Field Sources

Magnetic fields are produced by electric currents, which can be macroscopic

currents in wires, or microscopic currents associated with electrons in atomic

orbits.

Maxwell’s equations

mv

ev

ic

ic

qB

qD

JJt

DH

KKt

BE

Maxwell’s equations for TVF