Fundamental Concepts on Electromagnetic Theory

1. Department of Electrical & Electronic Engineering
Presentation on: Fundamental Concepts on Electromagnetic Theory.
Group: A
Name ID
Md:Sefat Ahmmad 17102931
Md:Razib Hossan 17102935
Md:Abdullah Rifat Khan 17102936
Md:Sadi Khan(Group leader) 17102940
AL-AMIN 16102904
Date: Sunday, September 23, 2018

2. Field:
If at every point of region, there is a corresponding value of some
physical function, the region is called field.
There are two types of field:
(i)Scalar field
(ii)Vector field
Scalar field: If the value of physical function at every point is scalar
quantity then the field is scalar field.
For example: temperature, density etc.
Vector field: If the value of physical function at every point is vector
quantity then the field is vector field. For vector field a number and a
direction are required.
For example: velocity, gravitational force etc.

3. Gradient:
The first type of multiplication is the multiplication of a vector by a
scalar and the result is a vector.
Physical interpretation of gradient:
let scalar function ∅ represents temperature, then 𝛻 ∙ ∅ or grad ∙ ∅
represent temperature gradient or rate of change of temperature with
distance. Thus although temperature is a scalar quantity having
magnitude only the temperature gradient 𝛻 ∙ ∅ is a vector quantity
having magnitude and direction both in the maximum rate of change of
temperature.
This gradient vector can be expressed:
𝛻 ∙ ∅ = 𝑖
𝜕∅
𝜕𝑥
+ 𝑗
𝜕∅
𝜕𝑦
+ 𝑘
𝜕∅
𝜕𝑧

4. Similarly if scalar function represents electric potential or
gravitational potential, then 𝛻 ∙ ∅ would represent electric
potential gradient or gradient of gravitational potential
respectively.

5. Md:Razib Hossen
ID:17102935

6. What is Divergence?
In vector calculus ,divergence is a operator that produces a scalar field ,given
the quantity of vector field’s source at each point.
More technically the divergence represents the volume density of the
outward flux of vector field.

7. What is curl?
• In vector calculus ,the curl is a vector operator that describes the
infinitesimal rotation of a vector field in three dimension Euclidean
space .
• A vector field whose curl is zero is called irrational curl .

8. What is co-ordinate system?
• A co-ordinate system is a system designed to establish position with respect to
given reference points .The co-ordinate system consists of one or more reference
points.

9. • There are three kinds of co-ordinate system . That’s are :
• Cartesian co-ordinate system(x, y ,z):That’s system species each
point in a plane by the pair of numerical co-ordinate .
• Cylindrical co-ordinate system(r, θ,z):That’s a three dimension co-
ordinates ,where each point is specified by the two polar co-
ordinate of its perpendicular projection onto same fixed point.
• Spherical co-ordinate system(r , , Ф ,):It is a co-ordinate system for
three dimensional space where position of the point is specified by
three numbers.

11. Md. Abdullah Rifat Khan
ID: 17102936

12. Static Field
Static Electric Field
Static Magnetic Field

13. Static Electric Field

14. Coulomb’s Law
The Force (F) between two charges (𝑄1 and 𝑄2) varies directly as the
product of the charges and inversely as the square of the distance (r)
between them.
𝐹 ∝
𝑄1 𝑄2
𝑟2 Or, 𝐹 = 𝑘
𝑄1 𝑄2
𝑟2 , k is proportionality constant.
𝑘 =
1
4𝜋𝜀
, 𝜀 is permittivity or dielectric constant of medium.
So the equation becomes, 𝐹 =
1
4𝜋𝜀
.
𝑄1 𝑄2
𝑟2
In vector form,
𝑭 =
1
4𝜋𝜀
.
𝑄1 𝑄2
𝑟2 𝒂 𝒓 , where 𝒂 𝒓 is unit vector.

15. Electric Field Intensity
Electric field intensity or electric field is denoted by E. If a unit charge
(q) is placed at any point near a fix charge (Q), the unit charge
experience a force. This force is know as Electric Field Intensity.
Applying coulomb’s law, 𝐹 =
1
4𝜋𝜀
.
𝑄 𝑞
𝑟2
Now 𝐸 =
𝐹
𝑞
=
𝑄 𝑞
𝑞.4𝜋𝜀𝑟2 =
𝑄
4𝜋𝜀𝑟2
In vector form, 𝑬 =
𝑄
4𝜋𝜀𝑟2 𝒂 𝒓

16. Electric Potential
If a small body having charge Q and a test charge q is moved from
infinity along radius line to point P at a distance R from the charge Q,
then work done (W) on the system,
𝑊 = −
∞
𝑅
𝐹. 𝑑𝑟
∞
𝑅
𝑄 𝑞
4𝜋𝜀𝑟2
Or, 𝑊 = − ∞
𝑅 𝑄 𝑞
4𝜋𝜀𝑟2 𝑑𝑟 = −
𝑄 𝑞
4𝜋𝜀
1
𝑟 ∞
𝑅
=
𝑄 𝑞
4𝜋𝜀𝑅
If the test charge is unit charge, then work done is potential V,
𝑉 =
𝑄 × 1
4𝜋𝜀𝑅
=
𝑄
4𝜋𝜀𝑅

17. Electric Charge Density 𝜌
The electric charge density (𝜌) is the ratio of total charge Q in a volume
V, to volume V.
So, 𝜌 =
𝑄
𝑉
Volume charge density, 𝜌 𝑣 = lim
∆𝑣→0
∆𝑄
∆𝑣
, unit 𝐶𝑚−3
Surface charge density, 𝜌𝑠 = lim
∆𝑠→0
∆𝑄
∆𝑠
, unit 𝐶𝑚−2
Linear charge density, 𝜌 𝐿 = lim
∆𝐿→0
∆𝑄
∆𝐿
, unit 𝐶𝑚−1

18. Static Magnetic Field

19. Faraday’s Law of Induction
If a varying magnetic field is placed near a conductor then an
electromotive force (emf) is induced across the conductor.
This emf is equal to the rate of change of the flux linkage of magnetic
field.
So, 𝑉 = −
𝑑𝜑
𝑑𝑡
Here, V = induced emf, in Volts
𝜑 = total magnetic flux, in weber

20. Biot-Savart’s Law
It deals with the magnetic field of current carrying element.
The magnetic flux density (dB), is directly proportional to the length (dl)
of the element, the current (I), the sine of the angle (𝜃) between
direction of the current and the vector joining a given point of the field
and the current element, and is inversely proportional to the square of
the distance (r) of the given point.
So, 𝑑𝐵 ∞
𝐼 𝑑𝑙 𝑠𝑖𝑛𝜃
𝑟2
Or, 𝑑𝐵 = 𝑘
𝐼 𝑑𝑙 𝑠𝑖𝑛𝜃
𝑟2 , 𝑘 is proportional constant
Here, 𝑘 =
𝜇
4𝜋
, 𝜇 is permeability
In vacuum, 𝜇 = 4𝜋 × 10−7
H/m

21. Biot-Savart’s Law
Now, 𝑑𝐵 =
𝜇
4𝜋
𝐼 𝑑𝑙 𝑠𝑖𝑛𝜃
𝑟2
Integrating the equation,
𝑑𝐵 =
𝜇
4𝜋
𝐼 𝑑𝑙 𝑠𝑖𝑛𝜃
𝑟2
So,
𝐵 =
𝜇𝐼
4𝜋
𝑠𝑖𝑛𝜃 𝑑𝑙
𝑟2
The unit of magnetic flux density is T.

22. Lenz’s Law
Lenz’s law states that when an emf is generated by a change in
magnetic flux according to Faraday’s Law, the polarity of the induced
emf is such, that it produces an current that’s magnetic field opposes
the change which produces it.
Case-1
Case-2

23. Lorentz Force
Lorentz force is the combination of electric force and magnetic force on
a point charge due to electromagnetic fields.
It is expressed mathematically in vector algebra as:
𝑭 = 𝑞𝑬 + 𝑞(𝒗 × 𝑩)
Here, F = force experience by the particle
q = charge of the particle
E = electric field
v = velocity of the particle
B = magnetic flux

24. Md : Sadi Khan
ID : 17102940

25. Current continuity equation
Displacement current

26. Current continuity equation:
A continuity equation is the mathematical way to express this kind of
statement.
For example: The continuity equation for electric charge states that
the amount of electric charge in any volume of space can only change
by the amount of electric current flowing into or out of that volume
through its boundaries

27. Let us consider a small volume element dv as shown in figure placed
inside a conducting medium. The current density 𝑗 a vector having the
same direction as the current flow. In general the current density 𝑗
have three rectangular components which vary with position.
∇ ∙ 𝑗=0
This equation is the differential equation involving 𝑗 at a point.
Continuous nature of 𝑗 and statement of Kirchhoff’s current law.
Total flow of current out of the volume must be equal to the negative
rate of charge with time. Total charge in the volume dv & 𝜌 is average
charge density.

28. 𝑠
𝑗 ∙ 𝑑𝑠 = −
𝜕
𝜕𝑡 𝑣
𝜌 dv
𝑠
𝑗 ∙ 𝑑𝑠= 𝑣
𝜕𝜌
𝜕𝑡
dv
Applying divergence theorem,
𝑣
∇ ∙ 𝑗 dv = 𝑣
𝜕𝜌
𝜕𝑡
∙ 𝑑𝑣 since ∇ ∙ 𝑗 =
𝜕𝜌
𝜕𝑡
This equation is valid for all volume v & is called as the general
equation of continuity between current density 𝑗 & charge density 𝜌 at
a point. This also called equation of conservation of charge.

29. Displacement current:
In electromagnetism, displacement current density is the quantity
𝜕𝐷
𝜕𝑡
appearing in Maxwell’s equations that is defined in terms of the rate
of change of D, the electric displacement field. Displacement current
density has the same units as electric current density, and it is a
source of the magnetic field just as actual current is. However it is not
an electric current of moving charges, but a time varying electric field.

30. Equation for displacement current:
The equation governing static electric field due to charge at rest and
static magnetic field due to steady currents are already derived and are
due to Gauss, Ampere & Faraday. They are reproduced here.
From Gauss’s Law,
From Faraday’s Law,

31. From Ampere’s Law,
Equation of continuity for steady current is given by,
Let us take divergence of both sides of (4),
Now, ∇ ∙ 𝑗 = -
𝜕𝜌
𝜕𝑡
⋯ ⋯ ⋯ ⋯(7)

32. From (7),
Thus although ∇ ∙ 𝑗 is not zero but the divergence of ( 𝑗 +
𝜕𝐷
𝜕𝑡
) is always
zero . Here , Maxwell made the assumption that the term 𝑗 of
equation (4) must replaced by ( 𝑗 +
𝜕𝐷
𝜕𝑡
) i.e.
∇x𝐻 = 𝑗 +
𝜕𝐷
𝜕𝑡

34. Department of Electrical & Electronic Engineering
Name: AL-AMIN
ID:16102904
Date: Sunday, September 23, 2018

35. 1.Definition & History of Maxwell’s equations:
Gauss Law
Gauss’ Magnetism Law
Faraday’s Law
Ampere’s Law
Maxwell’s equations in-
differential form
integral form
free space
Harmonically Varying fields

36. Maxwell’s equations
• Maxwell's equations describe how electric
charges and electric currents create electric and magnetic
fields. Further, they describe how an electric field can
generate a magnetic field, and vice versa.

37. Gauss Law
• Gauss law describes the nature of
electric field around electric charges.
The law is expressed in terms of electric
charge density and electric charge
density.
• The inverted triangle is called as the
divergence operator.
• The equations hold good at any point in
space. When the electric charge exists
any somewhere, the divergence of D at
that particular point is nonzero, else it is
zero.

38. Gauss’ Magnetism Law
• You can see that both the equations
indicate the divergence of the field. The
top equation states that the divergence of
the Electric flux density D equals the
volume of electric charge density.
• The second equation states the
divergence of the Magnetic Flux Density
(B) is null.

39. Faraday’s Law
Faraday was a scientist whose experiment setup led
to Faraday’s Law which is shown in the figure below.
• The experiment is not very complex.
When a battery is disconnected, no
electricity flows through the wire.
Hence, no magnetic flux is induced
in the iron (Magnetic Core). The iron
acts like a magnetic field that flows
easily in a magnetic material. The
purpose of the core is to form a path
for the flow of magnetic flux.

40. Ampere’s Law
• The law shows the relationship
between the flow of electric
current and the magnetic field
around it. Suppose the wire
carries a current I, the current
produces a magnetic field that
surrounds the wire.

41. Differential form

43. Where Q is the total charge within the volume v.
Or, simply
Or, simply
Integral form:

44. or
and
or

45. Thus ,summarizing ,we have

46. In Free space:
Differential Form:

47. Integral form:

48. Harmonically Varying Field:

49. Also,
Or,