ELECTROMAGNETIC FIELDS 1

Dr. Godfrey Mirondo Kibalya

kgmirondo@gmail.com
0773527950
 Vector Calculus
Definition:
A scalar is a quantity that has only magnitude
Quantities such as time, mass, distance,
temperature, entropy, electric potential, and
population are scalars.
A vector is a quantity that has both magnitude
and direction
Vector quantities include velocity, force,
displacement, and electric field intensity
Examples
Cross product
Coordinate Systems
Vector transformation
 Dell Operator
The Del operator, written V, is the vector differential operator.
In Cartesian coordinates,
This vector differential operator, otherwise known as the
gradient operator, is not a vector in itself, but when it operates
on a scalar function, for example, a vector ensues. The
operator is useful in defining,
1. The gradient of a scalar V, written, as
2. The divergence of a vector A, written as • A
3. The curl of a vector A, written as X A
4. The Laplacian of a scalar V, written as
 CURL OF A VECTOR AND STOKES THEOREM

The curl of A is an axial (or rotational) vector whose magnitude is the

maximum circulation of A per unit area as the area tends to zero and
whose direction is the normal direction of the area when the area is
oriented so as to make the circulation maximum.''
That’s

In Cartesian coordinates,
Which simplifies to

In cylindrical system

Which simplifies to
In spherical coordinates

Which simplifies to
Figure (a) shows that the curl of a vector field around P is
directed out of the page. Figure (b) shows a vector field
with zero curl
 Stoke’s theorem

Also, from the definition of the curl of A ,we may

expect that

Stokes's theorem 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 provided that A and
X A are continuous on S
Example: Determine the curl of the vector fields in the
previous example.
 LAPLACIAN OF A SCALAR
The Laplacian of a scalar field V, written as  2V
is the divergence of the gradient of V.
In Cartesian coordinates

Notice that the Laplacian of a scalar field is another scalar field.

In cylindrical coordinates

In spherical coordinates
 Classification of vector fields
A vector field is said to be solenoidal (Divergence less) if
. A  0
•Irrotational (potential/Conservative ) if

XA  0
Coulomb’s Law
Coulomb's law is an experimental law formulated in
1785 by the French colonel, Charles Augustin de
Coulomb.
It deals with the force a point charge exerts on another
point charge.
By a point charge we mean a charge that is located on a
body whose dimensions are much smaller than other
relevant dimensions. For example, a collection of electric
charges on a pinhead may be regarded as a point charge.
Charges are generally measured in coulombs (C). One
coulomb is approximately equivalent to 6 X 1018
electrons; it is a very large unit of charge because one
electron charge e = -1.6019 X 10-19C.
 Coulombs law states that the force F between two point
charges Q1 and Q2 is:
1. Along the line joining them
2. Directly proportional to the product QtQ2 of the charges
3. Inversely proportional to the square of the distance R
between them
Expressed mathematically,
1
k
4 o

where k is the proportionality constant.

 o is known as the permittivity of free space (in farads per meter)

and has the value
Substituting for K, we have

If point charges Q1 and Q2 are located at points having position

vectors r1 and r2, then the force F12 on Q2 due to Q1 as shown
below is given as,
Where
 Making this substitution, we have,

Or

Note that
If we have more than two point charges, we can use
the principle of superposition to determine the force on a
particular charge.
The principle states that if there are N charges
Q1,Q2,…….. QN located, respectively, at points with
position vectors r1; r2,. . ., rN, the
resultant force F on a charge Q located at point r is the
vector sum of the forces exerted on
Q by each of the charges Q1, Q2,. . ., QN. Hence:
 ELECTRIC FIELD INTENSITY,E
The electric field intensity (or electric field strength) K is
the force per unit charge when placed in the electric
field.
Thus

•The electric field intensity E is obviously in the direction of the force

F and is measured in newtons/coulomb or volts/meter.
•The electric field intensity at point r due to a point charge
located at r' is readily obtained as
Example
 ELECTRIC FIELDS DUE TO CONTINUOUS
CHARGE DISTRIBUTIONS
It is also possible to have continuous charge distribution
along a line, on a surface, or in a volume

It is customary to denote the line charge density,

surface charge density, and volume charge density by pL
(in C/m), ps (in C/m2), and pv (in C/m3), respectively.
The charge element dQ and the total charge Q due to
these charge distributions are obtained as,

(Line Charge)

The electric field intensity due to each of the charge

distributions pL, ps, and pv may be regarded as the
summation of the field contributed by the numerous
point charges making up the charge distribution
Thus by replacing Q with charge element dQ = pL dl, ps
dS, or pv dv and integrating, we get
 Line Charge distribution
Consider a line charge with uniform charge density pL extending
from A to B along the z-axis as shown in below. The charge
element dQ associated with element dl = dz of the line is
Example :
Electric flux density
Assignment to be presented on 4th
/11/12 2:00 pm
Assignment cont’d
 Assignment cont’d

End of Assignment
GAUSS’S LAW AND APPLICATION
ELECTRIC POTENTIAL
assignment
 Properties of materials
• Materials are classified according to their
conductivities as conductors and insulators or
technically as metals and insulators
• The conductivity of a material usually depends
on temperature and frequency. A material
with high conductivity ( a » 1) is referred to as
a metal whereas one with low conductivity (a
<< 1) is referred to as an insulator
 Properties of materials cont,d
• A material whose conductivity lies somewhere
between those of metals and insulators is
called a semiconductor
• The conductivity of metals generally increases
with decrease in temperature. At
temperatures near absolute zero (T = 0°K),
some conductors exhibit infinite conductivity
and are called superconductors e.g lead and
aluminum
 Convection and conduction currents
• Electric voltage (or potential difference) and
current are two fundamental quantities in
electrical engineering. Electric current is
generally caused by the motion of electric
charges.
• The current (in amperes) through a given area
is the electric charge passing through the area
per unit time.
• That’s
• Thus in a current of one ampere, charge is
being transferred at a rate of one columb per
second.
 Current Density,J
• We now introduce the concept of current
density J. If current I flows through a surface
, S the current density is

Or
Thus, the total current flowing through a surface
S is
• Depending on how I is produced, there are
different kinds of current densities:
• convection current density, conduction current
density, and displacement current density.
• We will consider convection and conduction
current densities here
• Convection current, as distinct from conduction
current, does not involve conductors and
consequently does not satisfy Ohm's law.
• It occurs when current flows through an
insulating medium such as liquid, or a vacuum. A
beam of electrons in a vacuum tube, for example,
is a convection current.
Consider a filament of Figure below. If there is a
flow of charge, of density pv, at velocity u = ay in y
direction,
the current through the filament is
or
Thus the conduction current density is

Or

The relationship in eq. above is known as the

point form of Ohm's law.
 Conductors
• A conductor has abundance of charge that is free
to move. Consider an isolated conductor, such as
shown in Figure 5.2(a). When an external electric
field Ee is applied, the positive free charges are
pushed along the same direction as the applied
field, while the negative free charges move in the
opposite direction. This charge migration takes
place very quickly.
• The free charges do two things. First, they
accumulate on the surface of the conductor and
form an induced surface charge.
Example
CONTINUITY EQUATION AND RELAXATION
TIME
• Due to the principle of charge conservation, the time
rate of decrease of charge within a given volume
must be equal to the net outward current flow
through the closed surface of the volume. Thus
current Iout coming out of the closed surface is
Having discussed the continuity equation and the
properties a and e of materials, it is
appropriate to consider the effect of introducing
charge at some interior point of a given
ELECTROSTATIC BOUNDARYVALUE
PROBLEMS
The procedure for determining the electric field E in the
preceding chapters has generally been using either Coulomb's
law or Gauss's law when the charge distribution is known, or
using E = -ϫV when the potential V is known throughout the
region. In most practical situations, however, neither the
charge distribution nor the potential distribution is known.
In this chapter, we shall consider practical electrostatic
problems where only electrostatic
conditions (charge and potential) at some boundaries are
known and it is desired to
find E and V throughout the region. Such problems are usually
tackled using Poisson's or
Laplace's equation or the method of images, and they are
usually referred to as boundary value problems.
 POISSON'S AND LAPLACE'S EQUATIONS

Poisson's and Laplace's equations are easily derived from

Gauss's law (for a linear material
medium)
UNIQUENESS THEOREM
Since there are several methods (analytical, graphical, numerical,
experimental, etc.) of solving a given problem, we may wonder
whether solving Laplace's equation in different ways gives
different solutions. Therefore, before we begin to solve Laplace's
equation, we should answer this question: If a solution of
Laplace's equation satisfies a given set of boundary conditions, is
this the only possible solution? The answer is yes: there is only
one solution. We say that the solution is unique. Thus any
solution of Laplace's equation which satisfies the same boundary
conditions must be the only solution regardless of the method
used. This is known as the uniqueness theorem. The theorem
applies to any solution of
Poisson's or Laplace's equation in a given region or closed
surface.
This is the uniqueness theorem: If a solution lo Laplace's
equation can be found that satisfies the boundary
conditions, then the solution is unique.

Before we begin to solve boundary-value problems, we

should bear in mind the three things that uniquely
describe a problem:
1. The appropriate differential equation (Laplace's or
Poisson's equation )
2. The solution region
3. The prescribed boundary conditions
A problem does not have a unique solution and cannot be
solved completely if any of the
three items is missing.
GENERAL PROCEDURE FOR SOLVING POISSON'S
OR LAPLACE'S EQUATION
 Steady magnetic fields
There are two major laws governing magneto static
fields:
(1) Biot-Savart's law and
(2) Ampere's circuit law.
Like Coulomb's law, Biot-Savart's law is the general
law of magneto- statics. Just as Gauss's law is a
special case of Coulomb's law, Ampere's law is a
special case of Biot-Savart's law and is easily
applied in problems involving symmetrical
current distribution.
BIOT-SAVART’S LAW
 BIOT- SAVART’S LAW
Biot-Savart's law states that the magnetic field intensity dH
produced at a point P, by the differential current element Idl is
proportional to the product Idl and the sine of the angle between
the element and the line joining P to the element and is inversely
proportional to the square of the distance R between P and the
elementThat’s
From the definition of cross product
 Different current distributions
If we define K as the surface current density (in
amperes/meter) and J as the volume current density
(in amperes/meter square), the source elements are
related as

Thus Biot-Savart law can be written as

Field due to filamentary conductor
We assume that the conductor is along the z-axis with
its upper and lower ends respectively subtending
angles and at P, the point at which H is to be
determined
Consider the contribution dH at P due to an element dl at p (0,
0, z),
As a special case, when the conductor is semi-infinite (with
respect to P) so that point A is now at (0, 0, 0) while B is at
; a1 = 90°, a2 = 0°,
then H becomes

Another special case is when the conductor is infinite in length.

For this case, point A is at
 Exercise
Q.1 The conducting triangular loop in figure below carries a current of 10 A. Find H at
(0, 0, 5) due to side 1 of the loop.
Example:
Find H at ( - 3 , 4, 0) due to the current filament shown in Figure below
Example:
 AMPERE’S CIRCUITAL LAW
Ampere's circuit law states that the line integral of the tangential
component of H around a closed path is the same as the net current
Ienc enclosed by the path.
In otherwords the circulation of H equals Ienc; that is,
Ampere's law is similar to Gauss's law and it is easily
applied to determine H when the current distribution
is symmetrical.
Maxwell’s equations
By applying Stokes's theorem to the left-hand side

But

By comparison

Thus the field H is not conservative

 Application of Ampere’s law
a) Infinite line current
Consider an infinitely long filamentary current I along the z-axis as
in Figure below. To determine H at an observation point P, we
allow a closed path pass through P. This path, on which Ampere's
law is to be applied, is known as an Amperian path (analogous to
the term Gaussian surface).
Since the path encloses the entire current, according to ampere’s
law, we have that
b) Infinite sheet of current

In general, for an infinite sheet of current density K

A/m,

where a is a unit normal vector directed from the

n

current sheet to the point of interest

 Magnetic flux density
In free space, the magnetic flux density B is related
to the magnetic field intensity H according to B=μH
Maxwell’s equations for static EM
Fields