ELECTROMAGNETIC FIELDS

Introduction to Electromagnetic
Fields
• Electromagnetics is the study of the effect of
charges at rest and charges in motion.
• Some special cases of electromagnetics:
– Electrostatics: charges at rest
– Magnetostatics: charges in steady motion (DC)
– Electromagnetic waves: waves excited by
charges in time-varying motion

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 Introduction to Electromagnetic
Fields
Fundamental laws of Maxwell’s
classical electromagnetics equations

Special Electro- Magneto- Electro- Geometric

cases statics statics magnetic Optics
waves
𝜕
Statics: ≡0
𝜕𝑡

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 MAXWELL’S EQUATIONS
• Laws of electricity and magnetism were established
in 1873 by James Clerk Maxwell (1831-1879).
• He suggested that if electromagnetic waves were to
be generated by a source, the waves can be
represented by a 3-dimensional vector notation when
observed from some point (r)r and time (t)t from the
source.

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• The
  H (r ,3-D
t )  D(vector
 
r , t )  J (r , t )  J notation
   

D
is given by;
t t

     B (The magnetomotive force around a closed path is equal to the

  E (r , t )   B ( r , t )   mi 
sum of t
conduction current density
t and displacement current density)

( Electromotive force around a closed path is equal to sum of the

  
time t )   (r , t )  of
. D(r , derivative  magnetic flux density and impressed magnetic flux current through the surface
bounded by the path).
 
. B (r , t )  0 (The total electric displacement density through a closed surface is
equal to the total charge within the volume).
    
E , B, H , D(The
, J andtotal
 magnetic flux merging from surface is zero)

Where are real functions of position and time

.
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  
E (r , t )
 = Electric field strength (volts/m)
H (r , t )

J (r , t ) = Magnetic field strength (amperes/m)
 
B(r , t )
 
= Electric current density
D(amperes/m²)
(r , t )

 (r , t )
mi = Magnetic flux density (webers/m²)
= Electric displacement (amperes/m²)
= Electric charge density
(coulombs/m³)
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 
NB. For .J  
continuity: t

(Electric current density J and electric charge density

ρ are governed by the Continuity law which states
that ‘The electric current and charge densities are
conserved)

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 ALTERNATIVE MAXWELL’S
EQUATIONS
B
 E    Kc  Ki
t
D
 H   Jc  Ji
t
  D  qev
  B  qmv

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 qev
J  
t Continuity equations

qmv
K  
t
Kc = magnetic conduction current density (V/m 2)
Ki = magnetic impressed current density (V/m 2)
qmv = magnetic charge density (Wb/m3)

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 Electrostatics
• Electrostatics is the branch of
electromagnetics dealing with the
effects of electric charges at rest.
• The fundamental law of electrostatics is
Coulomb’s law.
 Electric Charge
• Electrical phenomena caused by friction are part of our
everyday lives, and can be understood in terms of
electrical charge.
• The effects of electrical charge can be observed in the
attraction/repulsion of various objects when “charged.”
• Charge comes in two varieties called “positive” and
“negative.”
 Electric Charge
• Objects carrying a net positive charge attract those
carrying a net negative charge and repel those carrying a
net positive charge.
• Objects carrying a net negative charge attract those
carrying a net positive charge and repel those carrying a
net negative charge.
• On an atomic scale, electrons are negatively charged and
nuclei are positively charged.
 Electric Charge
• Electric charge is inherently quantized such that the
charge on any object is an integer multiple of the smallest
unit of charge which is the magnitude of the electron
charge e = 1.602  10-19 C.
• On the macroscopic level, we can assume that charge is
“continuous.”
 ELECTRIC CHARGE
• A parallel plate capacitor charged with one plate positive
and the other negative, will develop an electric field
between the plates.
• If an electron is placed between the plates, a force will act
on the electron tending to push it away from the negative
plate towards the positive plate.
• Similarly, a positive charge placed in the same field will
experience a force tending to push it away from the
positive plate towards the negative plate.
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 ELECTRIC CHARGE
• The region between the two plates in which an electric charge
experiences a force, is called an Electrostatic field .
• The direction of the field is the direction of the force on a positive charge
that is placed on the field.

A +++++ ++
+
B --------

Fig.1Electrostatic field

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 ELECTRIC CHARGE
• Such a field may be represented in magnitude and
direction by lines of electric force drawn between the
charged surfaces.
• The closeness of the lines is an indication of the field
strength.
• When a p.d is established between any two points, an
electric field will always be established.
+
+ _

a. Isolated point charge

b. Adjacent charges of opposite polarities

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Fig 2. Typical field 309 UZ
 ELECTRIC CHARGE
• Electric field lines of force (flux lines) are continuous
and start and finish on point charges.
• Also, field lines cannot cross each other.
• When a charged body is placed close to an
uncharged body, an induced charge of opposite sign
appears on the uncharged body.
• This induced charge is due to the field lines of a
charged body that terminate on the surface of the
uncharged body.
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 Coulomb’s Law
• Coulomb’s law is the “law of action” between charged
bodies.
• Coulomb’s law gives the electric force between two point
charges in an otherwise empty universe.
• A point charge is a charge that occupies a region of space
which is negligibly small compared to the distance
between the point charge and any other object.
 Coulomb’s Law
• The force of attraction or repulsion between two
electrically charged point charges or bodies is
proportional to the magnitude of the product of the two
charges and inversely proportional to the square of the
distance separating the charged bodies.
• where in air.
• If the unit vector in the direction of is , then we restate
Coulomb’s law as:

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 Coulomb’s Law
Q1
r12 Q2 Unit vector in
direction of R12
F 12
Q1 Q2
Force due to Q1 F 12  aˆ R1 2
acting on Q2 4  0 r12
2
 Coulomb’s Law
• The force on Q1 due to Q2 is equal in magnitude but
opposite in direction to the force on Q2 due to Q1.

F 21   F 12
 Electric Field
Qt
• Consider a point charge Q r
placed at the origin of a
coordinate system in an
Q
otherwise empty universe.
• A test charge Qt brought near Q QQt
experiences a force: F Qt  aˆ r
40 r 2
 Electric Field
• The existence of the force on Qt can be attributed to an
electric field produced by Q.
• The electric field strength produced by Q at a point in
space can be defined as the force per unit charge acting
on a test charge Qt placed at that point.

F Qt
E  lim
Qt  0 Q
t
 Electric Field
• The electric field describes the effect of a stationary charge on
other charges and is an abstract “action-at-a-distance”
concept, very similar to the concept of a gravity field.
• The basic units of electric field strength are newtons per
coulomb.
• In practice, we usually use volts per meter as a measure of
electric field strength E.
• In a simple parallel plate capacitor whose plate separation is r
and pd across the plates V, has an electric field strength given
as:
• volts/metre.
• In all other cases we define E as the force per unit charge.
 Electric Field
• For a point charge at the origin, the
electric field at any point is given by
_
Q Qr
E r   aˆ r 
4 0 r 2
4 0 r 3
 Electric Field
• For a point charge located at a point P’
described by a position vector r 
the electric field at P is given by :
P
_
_
QR
E r   r
R
4 0 R 3 P’
Q
where r
_
R  r  r O

R  r  r
 Electric Field
• In electromagnetics, it is very popular to describe the
source in terms of primed coordinates, and the observation
point in terms of unprimed coordinates.
• As we shall see, for continuous source distributions we
shall need to integrate over the source coordinates.
 Electric Field

• Using the principle of superposition, the

electric field at a point arising from multiple
point charges may be evaluated
¿ as:
𝑛
𝑄𝑘 𝑅𝑘
𝐸 ( 𝑟 )=∑ 3
𝑘=1 4 𝜋 𝜀0 𝑅 𝑘
 Continuous Distributions of Charge
• Charge can occur as
– point charges (C)
– volume charges (C/m3)
– surface charges (C/m2)
– line charges (C/m)
 Continuous Distributions of Charge

• Volume charge density

Qencl

r DV’

Qencl
qev r   lim
V  0  V 
 Continuous Distributions of Charge
• Electric field due to volume charge
density

r
dV’
Qencl V’
r P

_
qev r dv R
d E r  
4 0 R 3
Electric Field Due to Volume Charge
Density

_
1 qev r  R
E r    dv 
4 0 V  R 3
 Continuous Distributions of Charge

• Surface charge density

Qencl

r DS’

Qencl
qes r   lim
S   0  S 
Continuous Distributions of Charge
• Electric field due to surface charge
density

r
dS’
Qencl S’
r P

qes r ds R
d E r  
40 R 3
Electric Field Due to Surface Charge
Density

1 qes r  R
E r    ds 
40 S  R 3
 Continuous Distributions of Charge
• Line charge density

r DL’ Q
encl

Qencl
qel r   lim
L   0  L 
 Continuous Distributions of Charge
• Electric field due to line charge density

r
′
𝑑 𝑙 Qencl r P

𝑞 𝑒𝑙 ( 𝑟 ) 𝑑𝑙 𝑅
′ ′
𝑑 𝐸 (𝑟 )= 3
4 𝜋 𝜀0 𝑅
Electric Field Due to Line Charge Density

1 qel r  R
E r    d l 
40 L R 3
 Electrostatic Potential
• An electric field is a force field.
• If a body being acted on by a force is
moved from one point to another, then
work is done.
• The concept of scalar electric potential
provides a measure of the work done in
moving charged bodies in an
electrostatic field.
 Electrostatic Potential
• The work done in moving a test charge from
one point to another in a region of electric field:

F
b
a
q dl
b b
Wa b    F  d l  q  E  d l
a a
 Electrostatic Potential
• In evaluating line integrals, it is customary to take
the dl in the direction of increasing coordinate
value so that the manner in which the path of
integration is traversed is unambiguously
determined by the limits of integration.

b a
x
3 5 3
Wa b   q  E  aˆ x dx
5
 Electrostatic Potential
• The electrostatic field is conservative:
– The value of the line integral depends only
on the end points and is independent of the
path taken.
– The value of the line integral around any
closed path is zero.

C

C
E  d l  0
 Electrostatic Potential
• The work done per unit charge in
moving a test charge from point a to
point b is the electrostatic potential
difference between the two points:
b
Wa b
Vab    E  d l
q a
electrostatic potential difference
Units are volts.
 Electrostatic Potential
• Since the electrostatic field is
conservative we can write
b P0 b
Vab    E  d l    E  d l   E  d l
a a P0

b  a 
  E  dl    E  dl 
 
P0  P0 
 V b   V a 
 Electrostatic Potential
• Thus the electrostatic potential V is a scalar field that is
defined at every point in space.
• In particular the value of the electrostatic potential at any
point P is given by

P
V r     E  d l
P0
reference point
 Electrostatic Potential
• The reference point (P0) is where the potential is zero
(analogous to ground in a circuit).
• Often the reference is taken to be at infinity so that the
potential of a point in space is defined as

P
V r     E  d l

 Electrostatic Potential and Electric Field
• The work done in moving a point charge from point a to
point b can be written as

Wa b  Q Vab  QV b   V a 
b
 Q  E  d l
a
 Electrostatic Potential and Electric Field
• Along a short path of length Dl we have

W  QV  Q E  l
or
V   E  l
 Electrostatic Potential and Electric Field
• Along an incremental path of length dl we have
dV   E  d l
• From the definition of directional derivative:

dV  V  d l
 Electrostatic Potential and Electric Field
• Thus:

E  V

the “del” operator

 Gauss’s law
• The law states that the electric flux passing through any
enclosed surface is equal to the total charge enclosed by
that surface.
• .ds=charge enclosed=Q.
• The charge enclosed may be several point charges, in
which case:

Gauss’s law may be written in terms of charge distribution

as:
.ds=
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 Concentric Cable
• A concentric cable is a cable which contains more than
one conductors (cylinders).
• An example is a coaxial cable.
• The distribution of the electric field between the two
concentric cylinders is shown in figure below.

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• Consider a concentric cable whose inner conductor is of radius a
and outer conductor is of radius b as shown in figure below.

• The capacitance C, electric field strength E and inductance L of

such a cable are given as:

• Where r is the radius of an elementary

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 Tutorial Questions

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08/02/2024 EENG 309 UZ
• Q1. a. State Coulomb’s law of electrostatic force acting on two
point charges. [4 marks]
• b. Define electrostatic potential difference between any two points
and. [4 marks]
• c. Two point charges are at a distance apart along the z-axis as
shown in figure Q1 below. If each point charge is of value, what is
the total electric field strength at a point, equidistant from the two
point charges along the axis? [8 marks]
• d. What is the electric field at [4 marks]

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 z
1 q y

d 0 r
P
2 q x

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