Subject - Electromagnetic Fields

Year & Sem -B.E II year, Sem- III

Unit - I
Presented by – Dr G S Uthayakumar
Designation - Associate Professor
Department-Electronics and Communication Engineering

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 Unit I - Review of Vector Calculus

Scalars and vectors

• Scalars - A scalar is a quantity which is wholly characterized by it’s magnitude .Ex- temperature,
mass, volume, density, speed, electric Charge etc..

• Vector- A vector is a quantity which is characterized by both magnitude and direction .Ex- Force,
velocity, displacement, electric field intensity, acceleration etc.

• Scalar field- A field is a region in which a particular physical function has a value at each and
every point in that region. Ex- Temperature of atmosphere.

• Vector Field- If a quantity which is specified in a region to define a field , is a vector then
the corresponding field is called vector field Ex- wind velocity of atmosphere,
displacement of flying bird in a space.
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Representation of a vector
A vector can be represented by a
straight line with an arrow in a
plane
The length of the segment is
magnitude while the arrow indicates
the direction of the vector.
The vector showing is symbolically
denoted as OA.
OA = R
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 Unit Vector
A unit vector has a function to indicate the
direction, it’s magnitude is always unity.
• Consider a unit vector aOA in the direction of OA as
shown.
• This vector indicates the direction of OA but it’s
magnitude is unity.

• If a vector is known then the unit vector along that

vector can be obtain by dividing the vector by it’s
magnitude.

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 Vector algebra
The various mathematical operations can be performed with vectors such as:-
1) Scaling 2) Addition 3) Subtraction
1) Scaling of a vector
This is nothing but the multiplication by a scalar to vector. Such a multiplication
changes the magnitude but not it’s direction , when the scalar is positive.

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2) Addition of vectors
consider two coplanar vectors as shown
• The procedure is to move one of the two
vectors parallel to itself at the tip of the other
vector.
• By the addition of A and B the resultant C is
obtained.
Fig (a) coplanar vector

• The direction of C is from origin O to the tip

of the vector moved
• If B is moved parallel to itself at the tip of A,
we get same resultant C. Thus
Fig.(b) Addition of vectors

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Another method of performing the addition of vectors is parallelogram rule
• complete the parallelogram as shown . Then diagonal of the parallelogram
represents the addition of the two vectors.

Parallelogram rule for addition

• By using any of these two methods , no. of vectors can be added to obtain resultant

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• The following basic laws of algebra are obeyed by the vectors

α and β are scalars

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3) Subtraction of vectors
The subtraction of vectors can be obtained from the rules of addition. If B is to be
subtracted from A then based on addition it can be represented as

Identical vector
Two vectors are said to be identical if there difference is zero

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The co-ordinate system
To describe a vector accurately and to express a vector in terms of its components it
is necessary to have some reference directions. Such directions are represented in
terms of various co-ordinate systems. These are-
1) Cartesian or rectangular co-ordinate system
2) cylindrical co-ordinate system
3) Spherical co-ordinate system

1) Cartesian or rectangular co-ordinate system

This system has three co-ordinate axes represented as x , y and z which are mutually
perpendicular to each other. These three axes intersects at a common point called
origin of the system.
There are two types of such systems called
a) Right handed system and b) Left handed system.

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 • Right handed system
means if x axes is
rotated towards y axes
through a smaller angle,
then this rotation causes
the upward movement
of right handed screw in
the z axes. If the right
hand is used then the
thumb , forefinger and
middle finger may be
identified , respectively
as x, y and z axes.

(a) Right handed Cartesian co-ordinate system , (b) The location of point P (1,2,3) and Q(2,-2,1),
(c) The differential volume element, dx, dy and dz are in general independent differentials.
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• A point is located by giving its x, y and z co-ordinates.These are respectively the
distances from the origin to the intersection of a perpendicular dropped from the point
to the x,y and z axes.
• The alternative method is- to consider a point as being at the common intersection of
three surfaces, the planes x=constant, y= constant and z = constant. The constant
being the co-ordinate values of the point
• If we visualize three planes intersecting at a general point P (x,y,z),we may increase
each co-ordinate value by a differential amount and obtain three slightly displaced
planes intersecting at P’ (x+dx, y+dy, z+dz)
• The six planes define a rectangular parallelepiped whose volume is dv= dx dy dz.
• The surfaces has different areas ds of dx dy , dy dz and dz dx.
• Finally the distance dl from P to P’ is the diagonal of the parallelepiped and has the
length of

Point P’ is indicated , but point P is located the only invisible corner.

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• Unit vectors- three unit vectors are ax , ay and az .
• position or distance vectors are – consider a point P (x1,y1,z1) . The distance of
point P from origin is represented by position vector or radius vector.

The magnitude of this vector is given by

• Now if

Then the displacement from P to Q is represented by

This is the length of vector PQ

We can find a unit vector along the direction PQ as

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Differential elements in Cartesian co-ordinate system
dx= differential length in x direction
dy= differential length in y direction
dz = differential length in z direction
differential vector length is

distance from P to P’
Differential Volume
Differential surface area

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• Example 1- Obtain the unit vector in the direction from the origin towards the point
P(3,-3,-2).

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• Example-2 Two points A(2,2,1) and B(3,-4,2)are given in Cartesian
system. Obtain the vector from A to B and a unit vector directed from
A to B.

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• Example-3 Given A(3,-2,1), B(-3,-3,5) , C(2,6,-4) Find-
(i) Vector from A to C, (ii) unit vector from B to A, (iii) the distance from B to C and
(iv) The vector from A to midpoint of the straight line joining B to C.

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 Vector Multiplication
• Scalar or Dot Product
• Vector or Cross Product
Scalar or Dot Product – is defined as a product of magnitude of A
and B ,and the cosine of the smaller angle b/w them.

The result of such a Dot product is Scalar hence it is also called as

Scalar product.
Properties of Dot Product-
(i) If the two vectors are parallel to each other ie. θ = 0ₒ , then
cosθ = 1 thus
(ii) If two vectors are perpendicular to each other ie. θ = 90ₒ , then
cosθ = 0 thus
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(iii) The Dot product obeys commutative Law-
(iv) The Dot product obeys Distributive Law-
(v) If the Dot product of vector with itself is performed then
(vi)Consider the unit vectors ax , ay and az in Cartesian co-ordinate
system. All these vectors are mutually perpendicular to each other,
hence there Dot product is zero.

(vii) Any unit vectors dotted with itself is unity,

(viii) Consider two vectors in Cartesian co-ordinate system,

This product has nine scalar terms as dot product obeys distributive
law, but six terms out of nine will be zero involving the Dot product
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Application of Dot product-
(i) To determine angle b/w two vectors as -
(ii) To find a component of a vector in the given direction-

From fig. (a) we can obtain the component (scalar) of B in the direction specified by the
unit vector a as- B . a = │B│ │a│cosθBa = │B│cosθBa
This sign of component is positive if and negative whenever
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In order to obtain component vector of B in the direction of a , we simply multiply the
component (scalar) by a as illustrated by fig(b).
For example- the component of B in the direction of ax is B . ax = Bx and the component
vector is Bx ax or (B . ax ) ax
The Geometrical term projection is also used with the dot product. Thus B . a is the
projection of B in the direction of a .

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Example 5- given two vectors-
Find the dot product and angle between the two vectors.
Solution- The dot product is-

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• Example 6- Given vector field . Find this vector field at
P(2,3,1) and it’s projection on
Solution- The vector field at P
substituting co-ordinates of P in G

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Vector Product or Cross Product- is defined as a product of magnitude of A and B ,and
the sine of the smaller angle b/w them. It is denoted as A × B. This product is the vector
quantity hence the direction of A × B is perpendicular to the plane containing A and B ,
and is along that one of the two perpendicular which is in the direction of advance of a
right- handed screw as A is turned into B.

Where = unit vector perpendicular to the plane of A and

B in the direction decided by right handed screw rule.

Properties of Cross Product-

(i) The cross product is not commutative-
Reversing the order of vectors A and B results in a unit vector in opposite direction.

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(ii) The cross product is not associative, thus –
(iii) With respect to addition the cross product is distributive. Thus,
(iv) If the two vectors are parallel to each other ie. θ = 0ₒ , then sin θ = 0 thus
cross product of such two vectors is zero.
(v) If the Cross product of a vector with itself is performed then
(vi) Consider the unit vectors ax , ay and az in Cartesian co-ordinate system.
All these vectors are mutually perpendicular to each other, then

Hence,

But if the order of unit vectors is reversed, the result is negative of the
remaining third unit vector. Thus -

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This can be remembered by a circle indicating cyclic permutations of cross product of
unit vectors shown in fig

(vii) Cross product in determinant form-

Consider two vectors in Cartesian co-ordinate system,

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This result can be expressed in determinant form

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• Example 7 - Given two coplanar vectors
obtain the unit vector normal to the plane containing the vectors A and B .
Solution-

This is unit vector normal to the plane containing the vectors A and B .

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 Products of three vectors
• Scalar triple product
• Vector triple product
Scalar triple product- The scalar triple product of three vectors A, B and C is
mathematically defined as-
Thus if,

Then the scalar triple product is obtained by the determinant

The result of this product is scalar hence the product is called scalar triple product.
The cyclic order a, b, c is important.
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Characteristics of scalar triple product-
• This product represents the volume of a parallelepiped with edges A, B and C drawn
from the same origins as shown in fig.

• This product depends only on the cyclic order of ‘ a b c’ and not on the position of ·
and × in the product. If cyclic order is broken by permuting two of the vectors, the
sign is reversed.
• If two of the three vectors are equal then the result of the scalar triple product is zero.

• The scalar triple product is distributive.

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Vector triple product - The vector triple product of three vectors A, B and C is
mathematically defined as-
The rule can be remembered as ‘bac-cab’ rule. The above rule can easily be proved by
writing the Cartesian components of each term in equation. The position of the
brackets is very important.
Characteristics of vector triple product-
• It must be noted that in the vector triple product-
but
This is because is a scalar and multiplication by scalar to a vector is
commutative.
• From the basic definition we can write-

• But dot product is commutative hence and so on. Hence

the result of the vector triple product is a vector.

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Example-8- The three fields are given by - A= 2 ax - 3 az, B= 2 ax - ay + 2 az and
C= 2 ax -3 ay + az. Find scalar and vector triple product.
Solution-

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2) cylindrical co-ordinate system
• consider any point as the intersection of three mutually perpendicular surfaces.
• These surfaces are a circular cylinder (ρ =constant), a plane (ф=constant), and
another plane (z= constant).
• Three unit vectors must also be defined-
they are directed toward increasing coordinate values and are perpendicular to the
surface on which that coordinate value is constant (i.e., the unit vector ax is normal to
the plane x =constant and points toward larger values of x).
three unit vectors in cylindrical coordinates , are- aρ , aф and az
• The unit vector aρ at a point P(ρ1,ф1,z1) is directed radially outward, normal to the
cylindrical surface ρ= ρ1. It lies in the planes ф=ф1 and z=z1 .
• The unit vector aф is normal to the plane ф= ф1 , points in the direction of increasing ф,
lies in the plane z = z1 , and is tangent to the cylindrical surface ρ= ρ1 .
• The unit vector az is the same as the unit vector az of the cartesian coordinate system.
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• A differential volume element in cylindrical coordinates may be obtained by increasing
ρ,ф and z by the differential increments dρ, dф and dz.
The two cylinders of radius ρ and ρ+dρ , the two radial planes at angles ф and ф+dф ,
and the two ``horizontal'' planes at ``elevations'' z and dz. Now enclose a small volume,
as shown in Fig. having the shape of a truncated wedge.
As the volume element becomes very small, its shape approaches that of a rectangular
parallelepiped having sides of length dρ; dф and dz.
Note that dρ and dz are dimensionally lengths, but dф is not; ρdф is the length.
• The surfaces have areas of ρ dρ dф , dρ dz, and ρ dф dz.
• and the volume becomes ρ dρ dф dz:
• The variables of the rectangular and cylindrical coordinate systems are easily related to
each other - x= ρ cosф
y= ρ sinф
z=z
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• we may express the cylindrical variables in terms of x, y, and z:
and z=z
• Using the above equations, scalar functions given in one coordinate system are easily
transformed into the other system.
• A vector function in one coordinate system, requires two steps to transform it to another
coordinate system,
In cartesian coordinate system, Vector A= Ax ax + Ay ay + Az az
in cylindrical coordinates coordinates - A = Aρ aρ + Aф aф + Az az
To find any desired component of a vector, from the definition of the dot product –
Aρ = A • aρ and Aф = A • aф
Expending these products-
Aρ = (Ax ax + Ay ay + Az az) • aρ = Ax ax• aρ + Ay ay • aρ
Aф = (Ax ax + Ay ay + Az az ) • aф = Ax ax • aф + Ay ay • aф
Az = (Ax ax + Ay ay + Az az ) • az = Az az • az = Az
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• Dot products of unit vectors - ax• aρ , ay • aρ , ax • aф , ay • aф Can be determined
by the following table-

• aρ aф az

ax cosф -sinф 0
ay sinф cosф 0
az 0 0 1

Q. – (a) Give the Cartesian coordinates of the point C (ρ = 4.4, ф = -115⁰ , z = 2).
(b) Give the Cylindrical coordinates of the point D (x = -3.1, y = 2.6 , z = -3).
(c) specify the distance from C to D.
Ans- C (x = -1.860, y = 3.99 , z = 2) ; D (ρ = 4.05, ф = 140.0⁰ , z = -3); 8.36

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Example-9-Transform the vector field to cylindrical co-
ordinate system at point P(10,-8,6)
Solution- from the given field W,
Wx = 10, Wy = -8 and Wz = 6
Now Wρ = W•aρ = =
=
For point P, x= 10, and y= -8

So, Wρ =
Now

Hence, W = 12.804 aρ + 6 az in cylindrical system

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Example-10- Give the Cartesian coordinates of the vector
At point P(x=5, y= 2, z= -1)
Solution- the given vector is in cylindrical system

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3) Spherical co-ordinate system
• The surfaces which are used to define spherical co-ordinate system on
the three Cartesian axis are –
Sphere of radius r, origin as the centre of sphere
A right circular cone, with it’s apex at the origin and it’s axis as Z
axis. It’s half angle is θ. It rotates about Z axis and θ varies from 0 to
180⁰.
A half plane perpendicular to xy plane containing Z axis, making an
angle ф with the xy plane.
• The three co-ordinates of point P are (r, θ, ф)
• The ranges of the variables are-

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• The point P is defined as the intersection of three surfaces- r = a constant, θ = a constant
and ф = a constant.
• The intersection of a sphere r = constant and cone – θ = constant is a horizontal circle
whose radius is r sinθ.
• Now consider the intersection of ф = constant plane with r = constant and θ = constant
planes as shown in fig., this defines a point P.
• Three unit vectors may be defined at any point. Each unit vector is perpendicular to one of
the three mutually perpendicular surfaces and oriented in that direction in which the
coordinate increases.
• The unit vector is ar directed radially outward, normal to the sphere r = constant, and lies
in the cone θ = constant and the plane ф = constant.
• The unit vector aθ is normal to the conical surface, lies in the plane, and is tangent to the
sphere and oriented in the direction of increasing θ.
• The third unit vector aф is the same as in cylindrical coordinates, being normal to the
plane and tangent to both the cone and sphere. It is oriented in the direction of increasing
ф
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• A differential volume element may be constructed in spherical coordinates by increasing r,
θ , and ф by dr, dθ , and dф as shown in Fig.
• The distance between the two spherical surfaces of radius r and r + dr is dr; the distance
between the two cones having generating angles of θ and dθ is rdθ ; and the distance
between the two radial planes at angles ф and dф is found to be r sinθ dф.
• The surfaces have areas of r dr dθ , r sinθ dr dф , and r2 sinθ dθ dф.
• The volume is r2 sinθ dr dθ dф.
• The transformation of scalars from the cartesian to the spherical coordinate system is
easily made by using the relation:
x = r sinθ cos ф
y = r sinθ sin ф and z = r cos θ
• The transformation in the reverse direction is achieved with the help of

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• Differential elements in Spherical co-ordinate system

differential vector length is

and
Differential Volume
Differential surface areas are

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Relationship between Cartesian and spherical co-ordinate system

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• Dot products of unit vectors in spherical and Cartesian coordinate systems

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Example-11-
Slution-

•-

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Example-12-
Solution-

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• Distance in all co-ordinate systems-
Consider two points A and B with the position vectors as

Then the distance d between two points in all the three co-ordinate systems are
given by-

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• Transformation of vectors from spherical to cylindrical and
from cylindrical to Spherical system
Let the vector A in spherical system
The components of vector in cylindrical system is given by-

Now

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• Dot products of unit vectors - aρ• ar , aρ • aθ , aρ • aф , aф • aθ , , az • ar, , az • aθ ,
and , az • aф Can be determined by the following table-

• ar aθ aф
aρ sinθ cosθ 0
aф 0 0 1
az cosθ -sinθ 0

• Similarly a vector in cylindrical can easily be transferred into the Spherical

system

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Types of integral related to Electromagnetic Theory
A charge can exist in point , line , surface and volume form. Hence while dealing
with such charge distribution, the following types if integrals are required-
• Line integral
• Surface integral
• Volume integral
1) Line integral- a line can exist as a straight line or it can be a distance
travelled along a curve, thus in general from mathematical point of view, a line
is a curved path in space.
Consider a vector field F as shown in fig., the curved path shown in the field is
p-r . This is called the path of integration and corresponding integral can be
defined as-

Where dl = elementary length

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• This is called the line integral of F around the curved path L. It represents the tangential
component of F along the path L
• The curved path can be of two types-
(i) open path as p-r
(ii) closed path as p-q-r-s-p
• The closed path is also called Contour. The corresponding integral is called contour
integral, closed integral or circular integral. Mathematically-
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 if there exist a charge along a line as shown ,
then the total charge is obtained by calculating
a line integral.

where ρL = Line charge density (charge per

unit length (c/m))

2) Surface Integral – in electromagnetic theory, a charge may exist in a distributed form .

It may be spreaded over a surface as shown in fig(a). similarly a flux ф may pass through
a surface as shown in fig (b).
While doing analysis of such cases an integral is required is called surface integral, to be
carried out over a surface related to a vector field.

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• For charge distribution shown in fig (a), we can write for the total charge existing on
the surface as

• Where ρS = surface charge density in C/m2 ; dS= elementary surface

• Similarly for the fig.(b), the total flux crossing the surface S can be expressed as,

where
• Both the above equations represents the surface integrals and mathematically it
becomes a double integration while solving the problems.
• If the surface is closed, then it defines as a volume and corresponding surface
integration is given by,

• This represents the net outward flux of vector field F from surface S.

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3) Volume integral – if the charge distribution exists in a three dimensional volume form
as shown in fig. then a volume integral is required to calculate the total charge.
thus if ρv is the volume charge density over a volume V then the volume integral is
defined as-

Where dv= elementary volume

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Example -13 calculate the circulation of vector field,
Around the path L defined by- 0 ≤ r ≤ 3, 0 ≤ ф ≤ 45⁰ and Z = 0 as shown in fig.(a)

Solution- Divide the given path L into three sections.

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Harmonic field- A scalar field is said to be harmonic in a given region, if it’s
laplacian vanishes in the region.
Mathematically for a scalar field V to be harmonic,

This equation is called Laplace’s equation.

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Del operator- The Del operator , written , is the vector differential operator.
In Cartesian coordinates-

The operator is useful in defining-

1) The gradient of scalar V, written as V.
2) The divergence of a vector A, written as •A
3) The curl of a Vector A, written as, ×A
4) The Laplacian of a scalar V, written as V
Del operator in cylindrical coordinates-

Del operator in spherical coordinates-

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• Divergence- it is seen that gives the flux flowing across the surface S.
then mathematically Divergence is defined as the net outward flow of the flux per
unit volume over a close incremental surface. It is denoted as and is given
by -

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• Physically the divergence at a point indicate how much that vector field diverges from
that point.

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• Divergence Theorem- From the definition of divergence, we know that-

From this definition it can be written as- ……(i)

This equation (i) is known as the divergence theorem.

It states that- “The integral of the normal component of any vector field over a closed
surface is equal the integral of the divergence of this vector field throughout the
volume enclosed by that closed surface”.

The divergence theorem converts the surface integral into a volume integral, provided that
the closed surface encloses certain volume.
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• The theorem can be applied to any vector field but partial derivatives of that vector field
must exist. The divergence theorem as applied to flux density, both sides of the
divergence theorem give the net charge enclose by the closed surface( i.e. net flux
crossing the closed surface).

• Figure shows how closed surface S encloses a volume v for which divergence theorem
is applicable.

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Example- 14- A particular vector field is in cylindrical
system. Find the flux emanating due to this field from the closed surface of the cylinder
verify divergence theorem.
Solution- the outward flux is given by,

The surface made up of-

1) Top surface S1 for which Z=1, r varies from 0
to 4 4 and ф varies from 0 to 2π.
2) Lateral surface for which Z varies from 0 to 1 ,
ф from 0 to 2π and r = 4.
3) Bottom surface S3 for which Z=0, r varies
from 0 to 4 and ф varies from 0 to 2π

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 Gradient of a scalar- consider that in space let W be the unique function of x,y and z co-
ordinates in Cartesian system. This is the scalar function and denoted as W(x,y,z) .
Consider the vector operator in Cartesian system denoted as called del. It is defined
as,

• The operation of the vector operator del ( ) on a scalar function is called gradient of
scalar.

• Gradient of a scalar is a vector.

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• The gradient of a scalar W in various co-ordinate systems are given by-

• Properties of gradient of scalar-

1) The gradient W gives maximum rate of change of W per unit distance.
2) The W always indicates the direction of maximum rate of change of W.
3) The W at any point is perpendicular to the constant W surface, which passes
through the point.
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4) The directional derivative of W along the unit vector is (dot product),
which is the projection of W in the direction of .
If U is another scalar function then,
5)
6)
7)

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 Example 15- …in Cartesian
….. In cylindrical and ……in Spherical.
Find it’s gradient at P(0,1,1) for Cartesian, for cylindrical and P(3,60⁰,30⁰)
for spherical.
Solution-

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 Curl of a vector- The circulation of a vector field around a closed path is given by curl
of a vector. Mathematically it is defined as-

Where = area enclosed by the line integral in normal direction.

• Thus maximum circulation of per unit area as area tends to zero whose direction is
normal to the surface is called curl of .
Symbolically it is expressed as-
• Curl indicates the rotational property of vector field. If curl of vector is zero, the vector
field is irrigational.
• In various coordinate systems, the curl of vector is given by-

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Stock’s theorem – The Stock’s theorem relates the line integral to a surface integral.
It states that- “ The line integral of around a closed path L is equal to the integral
of curl of over the open surface S enclosed by the closed path L.”
Mathematically it is expressed as-

Where dL = perimeter of total surface S.

stock’s theorem is applicable only when
and are continuous on the surface S.
The path L and the open surface S enclosed
by path L for which stock’s theorem is
applicable are shown in fig.

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Example 16-

Solution-

The L.H.S. is already evaluated in previous

example 13 , which is 2.636.
To evaluate R.H.S. , find

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 Laplacian of a Scalar- The divergence of a vector and gradient of a scalar is discussed
earlier. The composite operator of these two is called Laplacian of a scalar.
If V is a scalar field then, the Laplacian of scalar V is denoted as and
mathematically defined as the divergence of the gradient of V.
The operator is called laplacian operator.
• In Cartesian coordinate system-

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Example-17

Solution-

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REFERENCES
• Engineering Electromagnetic by william H. Hayt, Jr. John A. Buck
• Principals of Electromagnetics by Matthew N. O. Sadiku.
• CBS problems and solution series ( Problems and solution of
Engineering Electromagnetics).
• https://nptel.ac.in/courses/108/106/108106073/
• https://nptel.ac.in/courses/117/103/117103065/

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