ELECTROMAGNETIC THEORY
B.E. 402
Unit 1
Review of vector calculus: orthogonal coordinate systems, gradient, divergence and curl. Laplacian operator for
scalar and vectors. Vector integral and differential identities and theorem
Static electric fields, Columb’s law, electric flux density and electric field intensity, permittivity, dielectric constant,
field of distributed charges in free space, potential function, Laplace’s and Poisson’s equations, electric dipole, stored
electric energy density. Boundary conditions at abrupt discontinuities between two media including conducting
boundaries, surface charge distribution capacitance between two isolated conductors
Basic Concept of Vectors
In engineering much of the work in both analysis and design involves forces. You will be familiar
with forces in structural members in space frames, and know that a force acts in a given
direction with a given magnitude. A force is a three dimensional quality and one of the most
common examples of a vector. Two other common vector quantities are acceleration and velocity.
In this section of the module we will introduce some formal mathematical notation and rules for
the manipulation (multiplication, addition etc.) of these vector quantities. In the early stages it
will be easy to recognise the geometric meaning of vector addition and multiplication, but as the
problems become more complex (notably, when they are three-dimensional) the formal rules
A Scalar quantity is one that is defined by a single number with appropriate units.
Some examples are length, area, volume, mass and time.
A Vector quantity is defined completely when we know both its magnitude (with units) and its
direction of application.
Some examples are force, velocity and acceleration.
Two very simple (and common) examples demonstrating the differences between scalars and
vectors are speed - a scalar, and velocity - a vector.
A speed of 10 km/h is a scalar quantity
A velocity of 10 km/h at 20 degrees is a vector quantity
Magnitude (or Modulus)
The magnitude, modulus or length of the vector a is written as a or OA and given in component form by
a = (a12 + a22 + a32 )1/ 2
The Unit Vector
A vector whose modulus is 1 is called a unit vector, sometimes written as aˆ
a = a / |a|
Vector Addition
The boat problem
A boat steams at 4 knots due East for one hour. The tide is running North-North-East at 3 knots. Where will the boat be after
one hour?
A diagram of the vectors involved looks like that below, where a represents the velocity of the boat and b represents the
velocity of the tide.
NNE
North
B
C
a+b
b
O
b
a
A
East
The net velocity of the boat is represented by the line OC which is the sum of a and b.
Scalar product
The scalar product of two vectors a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) is written with a dot (⋅) between the to vectors.
It is defined in component form as
a ⋅b = a1b1 +a2 b2 + a3b3
And in geometrical form
a ⋅b = b acosθ
Where θ is the angle between the two vectors and 0 ≤θ ≤ π .
The Vector (or Cross) Product
The main practical use of the vector product is to calculate the moment of a force in three dimensions. It is of very
limited use in two dimensions.
For two vectors a and b, the vector product is defined as:
(a ×b) = a b sinθnˆ
where θ is the angle between the vectors a and b, ( 0 ≤θ ≤ π ) and nˆ is the unit vector normal to both a and b.
Question 1 :
Define Rectangular Co ordinate system (rgpv May/june 2007,08,2011 )
. This point of intersection is called the origin and is denoted by the symbol O. The horizontal and vertical number lines are
called the x-axis and the y-axis, respectively. These axes divide the plane into four regions, called quadrants, which are
numbered as shown in FIGURE As we can see in Figure the scales on the x- and y-axes need not be the same. Throughout this
text, if tick marks are not labeled on the coordinates axes, as in Figure then you may assume that one tick corresponds to one
unit. A plane containing a rectangular coordinate system is called an xy-plane, a coordinate plane,
or simply 2-space
Question 2 :
Define cylindrical Co ordinate system (rgpv May/june 2005,09 )
Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height ( ) axis. Unfortunately,
there are a number of different notations used for the other two coordinates. Either
the azimuthal coordinates. Arfken (1985), for instance, uses
used.
or
is used to refer to the radial coordinate and either
, while Beyer (1987) uses
. In this work, the notation
or
to
is
Question 3 :
Define Spherical Co ordinate system (rgpv May/june 2010,13 )
Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for
describing positions on a sphere or spheroid. Define
when referred to as the longitude),
to be the azimuthal angle in the
-plane from the x-axis with
to be the polar angle (also known as the zenith angle and colatitude, with
the latitude) from the positive z-axis with
, and
(denoted
where
is
to be distance (radius) from a point to the origin. This is the convention commonly used
in mathematics.In this work, following the mathematics convention, the symbols for the radial, azimuth, and zenith angle coordinates are taken as
, and
, respectively. Note that this definition provides a logical extension of the usual polar coordinates notation, with
the
-plane and
,
remaining theangle in
becoming the angle out of that plane. The sole exception to this convention in this work is in spherical harmonics, where the
convention used in the physics literature is retained (resulting, it is hoped, in a bit less confusion than a foolish rigorous consistency might
engender).Unfortunately, the convention in which the symbols
and
are reversed (both in meaning and in order listed) is also frequently used,
especially in physics. This is especially confusing since the identical notation
but (radial, polar, azimuthal) to a physicist.
typically means (radial, azimuthal, polar) to a mathematician
Relations between Cartesian, Cylindrical, and Spherical Coordinate1
Figure : Standard relations between cartesian, cylindrical, and spherical coordinate systems. The
origin is the same for all three. The positive z-axes of the cartesian and cylindrical systems
coincide with the positive polar axis of the spherical system. The initial rays of the cylindrical
and spherical systems coincide with the positive x-axis of the cartesian system, and the rays
=90° coincide with the positive y-axis.Then the cartesian coordinates (x,y,z), the cylindrical
coordinates (r, ,z), and the spherical coordinates ( , , ) of a point are related as follows:
Question 4 Express Coulomb’s Law and Electric Field Intensity Permittivity ,Dielectric constant
( rgpv May june 2012.13 )
Coulomb's law states that:
The magnitude of the electrostatic force of interaction between two point charges is directly proportional to the
scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distance between
them.
The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force
between them is repulsive; if they have different sign, the force between them is attractive.
Coulomb's law can also be stated as a simple mathematical expression. The scalar and vector forms of the
mathematical equation are
and
where
respectively,
is Coulomb's constant (
Electric field
If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them
is attractive.
An electric field is a vector field that associates to each point in space the Coulomb force experienced by a test charge. In
the simplest case, the field is considered to be generated solely by a single source point charge. The strength and direction
of the Coulomb force
that
on a test charge
depends on the electric field
. If the field is generated by a positive source point charge
that it finds itself in, such
, the direction of the electric field points
along lines directed radially outwards from it, i.e. in the direction that a positive point test charge
would move if placed in
the field. For a negative point source charge, the direction is radially inwards.
The magnitude of the electric field
can be derived from Coulomb's law. By choosing one of the point charges to be the
source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the electric field
by a single source point charge
at a certain distance from it
.
in vacuum is given by:
created
Permittivity : permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium.
The permittivity of a medium describes how much electric field (more correctly, flux) is 'generated' per unit charge
in that medium. The vacuum permittivity ε0 (also called permittivity of free space or the electric constant) is the
ratio D/E in free space It also appears in the Coulomb force constant, ke = 1/(4πε0).
Its value is
where c0 is the speed of light in free space.
Dielectric constant: property of an electrical insulating material (a dielectric) equal to the ratio of
the capacitance of a capacitor filled with the given material to the capacitance of an identical capacitor in
a vacuum without the dielectric material. The insertion of a dielectric between the plates of, say, a parallelplate capacitor always increases its capacitance, or ability to store opposite charges on each plate,
compared with this ability when the plates are separated by a vacuum. If C is the value of the capacitance
of a capacitor filled with a given dielectric and C0 is the capacitance of an identical capacitor in a vacuum,
the dielectric constant, symbolized by the Greek letter kappa, κ, is simply expressed as κ = C/C0. Dielectric
constant is a number without dimensions. It denotes a large-scale property of dielectrics without specifying
the electrical behaviour on the atomic scale.
Material Dielectric constant Dielectric Strength kV/mm
Vacuum1.00000 kV/mm
Air (dry)1.000593 kV/mm
Polystyrene2.624 kV/mm
Paper3.616 kV/mm
Water80- kV/mm
Electric Potential :
An electric potential (also called the electric field potential or the electrostatic potential) is the amount of electric potential
energy that a unitary point electric charge would have if located at any point of space, and is equal to the work done by
anelectric field in carrying a unit positive charge from infinity to that point.This value can be calculated in either a static (timeinvariant) or a dynamic (varying with time) electric field at a specific time in units of joules per coulomb ( J C−1), or volts (V).
The electric potential at infinity is assumed to be zero
LaPlace's and Poisson's Equations
A useful approach to the calculation of electric potentials is to relate that potential to the charge
density which gives rise to it. The electric field is related to the charge density by the divergence
relationship
and 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
This mathematical operation, the divergence of the gradient of a function, is called
the LaPlacian. Expressing the LaPlacian in different coordinate systems to take advantage of the
symmetry of a charge distribution helps in the solution for the electric potential V. For example,
if the charge distribution has spherical symmetry, you use the LaPlacian in spherical polar
coordinates.
Since the potential is a scalar function, this approach has advantages over trying to calculate the
electric field directly. Once the potential has been calculated, the electric field can be computed
by taking the gradient of the potential.
Electric dipole
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a
system of electric charges, that is, a measure of the charge system's overall polarity. The SI units are Coulomb-meter (C m).
This article is limited to static phenomena, and does not describe time-dependent or dynamic polarization. The magnitude of
dipole moment determines the electric field strength. In the simple case of two point charges, one with charge +q and the
other one with charge −q, the electric dipole moment p is:
where d is the displacement vector pointing from the negative charge to the positive charge. Thus, the electric dipole
moment vector p points from the negative charge to the positive charge. An idealization of this two-charge system is the electrical
point dipole consisting of two (infinite) charges only infinitesimally separated, but with a finite p.
Boundary conditions for
and
When the space surrounding a set of charges contains dielectric material of non-uniform dielectric
constant then the electric field no longer has the same functional form as in vacuum. Suppose
forexample, that the space is occupied by two dielectric media whose uniform dielectric constants
are
and
. What are the boundary conditions on
and
at the interface between the two
media? Imagine a Gaussian pill-box enclosing part of the interface. The thickness of the pill-box
is allowed to tend towards zero, so that the only contribution to the outward flux of
comes
from the flat faces of the box, which are parallel to the interface. Assuming that there is no free
charge inside the pill-box (which is reasonable in the limit in which the volume of the box tends
to zero), then Eq. yields
where
is the component of the electric displacement in medium 1 which is
normal to the interface, etc. If the fields and charges are non time-varying then the
differential form of Faraday's law yield
, which gives the familiar
boundary condition (obtained by integrating around a small loop which straddles the
interface)
Generally, there is a bound charge sheet on the interface whose density follows from
Guess Law
In conclusion, the normal component of the electric displacement, and the tangential
component of the electric field, are both continuous across any interface Between
two mediums
Surface charge Distribution
Surface charge is the electrical potential difference between the inner and the outer surface of the different aggregate state:
liquid and gas, liquid and solid or gas and liquid. There are many different processes which can lead to a surface being
charged, including adsorption of ions, protonation/deprotonation, and the application of an external electric field. Surface
charge causes a particle to emit an electric field, which causes particle repulsions and attractions, and is responsible for
many colloidal properties. Surface charge practically always appears on the object surface when it is placed into a fluid All
fluids contain ions, positive cations and negative anions. These ions interact with the object surface. This interaction might
lead to the absorpation of some of them on the surface. If the number of adsorbed cations exceeds the number of adsorbed
anions, the surface would gain total positive electric charge
Capacitor
Capacitance is typified by a parallel plate
arrangement and is defined in terms of charge
storage:
where


Q = magnitude ofcharge stored on each
plate.
V = voltageapplied to the plates.
Spherical Capacitor
The capacitance for spherical or cylindricalconductors can be obtained by
evaluating the voltage difference between the conductors for a given
charge on each. By applying Gauss' law to an charged conducting sphere,
the electric field outside it is found to be
The voltage between the spheres can be found by integrating the electric field along a
radial line:
From the definition of capacitance, the capacitance is
Capacitor Combinations
Capacitors in parallel add ...
If
=
then
,
=
=
Capacitors in series combine as reciprocals ...
=
Charge on Series Capacitors
Since charge cannot be added or taken away from the conductor between series capacitors, the net
charge there remains zero. As can be seen from the diagram, that constrains the charge on the two
capacitors to be the same in a DC situation. This charge Q is the charge you get by calculating the
equivalent capacitance of the series combination and multiplying it by the applied voltage V.
You store less charge on series capacitors than you would on either one of them alone with the same
voltage!
Does it ever make sense to put capacitors in series? You get less capacitance and less charge storage than with either alone.
It is sometimes done in electronics practice
because capacitors have maximum working voltages, and with two "600 volt maximum" capacitors in series, you can increase the
working voltage to 1200 volts.
For further Reading
 Matthew N.O. Sadiku
 Hayt