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Vector Algebra - An Introduction - Field Theory.

Most of the physical quantities are either scalar or vector quantities.

SCALAR QUANTITY:
- Scalar is a number that defines magnitude. Hence a scalar quantity is defined as a quantity that has magnitude only.

- A scalar quantity does not point to any direction i.e. a scalar quantity has no directional component.
For example when we say, the temperature of the room is 30o C, we don’t specify the direction.

- Hence examples of scalar quantities are mass, temperature, volume, speed etc.

- A scalar quantity is represented simply by a letter – A, B, T, V, S.


VECTOR QUANTITY:

- A Vector has both a magnitude and a direction. Hence a vector quantity is a quantity that has both magnitude and direction.

- Examples of vector quantities are force, displacement, velocity, etc.

- A vector quantity is represented by a letter with an arrow over it.


UNIT VECTORS (aA):

- When a simple vector is divided by its own magnitude, a new vector is created known as the unit vector
Mathematically,  aA = A / |A|

- …

ElectroStatics - Questions With Answers (Short)

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1. State Stokes Theorem.

Answer: The line integral of a vector around a closed path is equal to the surface integral of the normal component of its curl over any surface bounded by the path

∫ H.dl = ∫ ∫ ( ∆ x H ) ds
where, H is the Magnetic field intensity

2. State The Condition For The Vector F To Be Solenoidal.

Answer:
∆ . F = 0
where, F = A i + B i + C i

3. State The Condition For The Vector F To Be Irrotational.

Answer:

∆ x F = 0 where, F = A i + B i + C i

4. Give The Relationship Between Potential Gradient and Electric Field.

Answer: 
E = - ∆V
where, E represents Electric Field Intensity and V represents Electric Potential

5. What Is The Physical Significance Of div D ?

Answer: The divergence of a vector flux density is electric flux per unit volume leaving a small volume. This is equal to the volume charge density.

6. What Are The Sources Of Electric Field & Magnetic Field?

Answer: Stationary charges produce electric field that are constant in time, hence the term electrostatics. Movin…

State The Difference Between Conduction & Displacement Current?

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The differences between Conduction Current and displacement current are:

1. Conduction current obeys ohm's law as V = (I / R) but displacement current does not obey ohm's law.

2. Conduction current density is represented by

 whereas displacement current density is given by

3. Conduction current is the actual current whereas displacement current is the apparent current produced by time varying electric field.

Antenna Theory - Solved Numerical's / Problems - ElectroMagnetic Theory...

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1) An electric field strength of 10 µV/m is to be measured at an observation point θ = π/2, 500 km from a λ/4 monopole antenna operating in air at 50 MHz.
(a) What is the length of the dipole?(b) Calculate the current that must be fed to the antenna.(c) Find the average power radiated by the antenna.(d) If a transmission line with Zo = 75 Ω is connected to the antenna, determine the standing wave ratio. --- For Solution CLICK HERE.

2) Calculate the directivity of
The Hertzian Monopole.The quarter-wave Monopole.  --- For Solution CLICK HERE.
3) A certain antenna with an efficiency of 95% has maximum radiation intensity of 0.5 W/sr. Calculate its directivity when
(a) The input power is 0.4 W
(b) The radiated power is 0.3 W   --- For Solution CLICK HERE.


4) Evaluate the directivity of an antenna with normalized radiation intensity

 --- For Solution CLICK HERE.

5) Determine the maximum effective area of a Hertzian dipole of length 10 cm operating at 10 MHz. If the antenna receives 3 µW of power, w…

Vector Analysis & Electrostatics - Index / Sitemap.

INDEX
Vector analysis is a mathematical tool with which electromagnetic (EM) concepts are most conveniently expressed. It's important to learn its rules and techniques first applying it.

VECTOR ALGEBRA:

- Scalars and Vectors.

- Unit Vectors.

- Position and Distance Vectors.

- Vector Multiplication.

- Components Of a Vector.

- Numericals / Solved Examples.


COORDINATE SYSTEMS & TRANSFORMATION:

In general, the physical quantities in ElectroMagnetics are functions of space and time. In order to describe the spatial variations of the quantities, its important to define all points uniquely in space in a suitable manner. This requires using an appropriate coordinate system. Hence its very important to understand the coordinate system first. 

- Introduction To Coordinate System.

- Cartesian Coordinate System / Rectangular Coordinate System (x, y, z).

- Differential Analysis Of Cartesian Coordinate System.

- Circular Cylindrical Coordinate System (ρ, φ, z).

- Differential Analysis Of Cylindrical Coo…

Transmission Lines - Solved Numericals / Problems - 2.

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Q.4) A 70 Ω lossless line has s = 1.6 and θΓ = 300o. If the line is 0.6λ long, obtain
(a) Γ, ZL, Zin
(b) The distance of the first minimum voltage from the load

Answer:

a) Using the smith chart, locate S at s = 1.6. Draw a circle of radius OS. Locate P where θΓ = 300o . At P,

| Γ | = OP / OQ = 2.1 cm / 9.2 cm = 0.228

Γ = 0.228 ∠300o

Also at P, ZL = 1.15 – j0.48,

ZL = Zo ZL = 70 (1.15 – j0.48) = 80.5 – j33.6 Ω

l = 0.6 λ = 0.6 x 720o = 432o = 360o + 72o

From P, move 432o to R. At R, Zin = 0.68 – j0.25

Zin = Zo Zin = 70 (0.68 – j0.25) = 47.6 – j17.5 Ω

b) The maximum voltage (the only one) occurs at θΓ = 180o ; its distance from the load is
     (180 – 60) λ / 720  =   λ / 6  =   0.1667 Ω




Q.5) A lossless 60 Ω line is terminated by a 60 + j60 Ω load.
(a) Find Γ and s. If Zin = 120 - j60 Ω, how far (in terms of wavelengths) is the load from the generator? Solve this without using the Smith chart.
(b) Solve the problem in (a) using the Smith chart. Calculate Zmax and Zin,min .…

Solved Exercise/Numericals - Transmission Lines - 1.

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Q.1) A transmission line operating at 500 MHz has Zo = 80Ω, α = 0.04Np/m, β = 1.5 rad/m. Find the line parameters R, L, G and C.

Answer:

Since Zo is real & α ǂ 0, this is a distortionless line.



Ro = α Zo = 0.04 x 80 = 3.2 Ω / m

G = α / Zo = 0.04 / 80 = 5 x 10-4 Ω / m

L = β Zo / ω = 1.5 x 80 / (2 π x 5 x 108 ) = 38.2 nH / m

C = L G / R = 38.2 x 10-9 x 5 x 10-4 / 3.2 = 5.97 pF / m



Q.2) A telephone line R = 30 Ω /km, L = 100 mH/km, G = 0 , and C = 20 µF/km. At f = 1 kHz, obtain:

a) The characteristics impedance of the line.
b) The propagation constant.
c) The phase velocity.


Answer:





Q.3) A 40 m long transmission line has Vg = 15 ∠0o Vrms , Zo = 30 + j60 Ω, and VL = 5 ∠-48o Vrms . If the line is matched to the load, calculate:

a) The input impedance Zin
b) The sending end current Iin and voltage Vin
c) The propagation constant γ



Answer:

a) Zg  =  Z1  --> Zin  =  Zo  =  30 + j60 Ω

b) Vin  =  Vo  =  ( Zin / (Zin + Zo) ) Vg  =  Vg / 2  =  7.5∠0oVrms

     Iin  =  Ip  =  …

SOLVED Numerical's - Antenna Theory - Page 7.

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19) At the far field, an antenna produces





Calculate the directive gain and the directivity of the antenna?

SOLUTION:





















20) For a thin dipole λ/16 long, find the
a) Directive gain
b) Directivity
c) Effective area
d) Radiation resistance

SOLUTION:


a)

























On substituting we get,

Gφ = 1.5 sin2θ

b) Directivity, D = Gφ, max = 1.5

c) Effective Area, Ae = (λ2 / 4π) Gφ = (1.5 λ2 sin2θ) / 4π

d) Radiation Resistance Rrad




SOLVED Numerical's - Antenna Theory - Page 6.

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16) Sketch the normalized E-field and H-field patterns for
a. A half-wave dipole
b. A quarter-wave monopole.
SOLUTION:

a)



b) The same as for λ/2 dipole except that the fields are zero for θ > π/2 as shown:





17) In free space, an antenna has a far zone field given by









Determine the radiated power?


SOLUTION:








































18) At the far field, the electric field produced by antenna is






Sketch the vertical pattern of the antenna. Your plot should include as many points as possible.
SOLUTION:


       f(θ) = | cosθ cosφ |

            For the vertical pattern, φ = 0 which means,

                       f(θ) = | cosθ | which is sketched below