Electromagnetic Wave Theory

1. Series: EMF Theory
Lecture: # 0.11
Dr R S Rao
Professor, ECE
Scalar and vector products, Del, gradient, divergence, curl.
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2. The dot and cross products between two arbitrary vectors A and
B are
• (dot) → scalar/dot product between two vectors
→ divergence of a vector
cos
AB 

A B
ˆ
sin
AB 
 
A B n
× (cross) →vector/cross product between two vectors
→ Curl of a vector
Electromagnetic
Fields
Vector
Calculus-I
2
Vector Algebra
Dot product can be found as product of magnitude of one vector
with component of the other vector along the first vector.

3. Vector Algebra
1
cos
A B
AB
  

•The angle a between two arbitrary vectors A and B
ˆ
sin
A B A B
n
A B AB 
 
 

•The unit normal to two arbitrary vectors A and B is
Electromagnetic
Fields
Vector
Calculus-I
3
Divergence and curl operations require application of del
operator over vector functions, through dot and cross.

4. Vector Algebra
     
A B C =B• C A =C• A B
    
     
   
A B C B A C C A B
•Triple products are
•The first product, formed with three arbitrary vectors, is a scalar and hence
called scalar triple product. A geometrical interpretation is, its absolute value
gives the volume of the parallelepiped, having the three vectors as three of its
contiguous edges.
•The second product, called triple cross product, results in a vector. The
expansion is sometimes called BAC CAB rule, useful to remember the
expression. Note that A× (B × C)is not in general equal to (A× B) × C.
Electromagnetic
Fields
Vector
Calculus-I
4

5. ˆ ˆ ˆ
x y z
x y z
  
  
  
Electromagnetic
Fields
Vector
Calculus-I
5
Del Operator
Also called Nabla, or Atled (delta spelled backwards), with symbol, s, it was
introduced in 1853 by William Rowan Hamilton (1805-1865) a mathematician and a
polymath.
• It is a three dimensional, partial differential vector operator.
• Even though defined in Cartesian system, it can be mapped into other co-ordinate
systems also. Its units are per meter.
• It can be applied over vector as well scalar functions.
• When applied over a vector function, it may give a scalar or a vector function.
• When applied over a scalar function, it results in a vector function.

6. 2 2 2
2
2 2 2
x y z
  
  
  
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Fields
Vector
Calculus-I
6
Laplacian Operator
• It is a three dimensional, second order, partial differential scalar operator.
• Even though defined in Cartesian system, it can be mapped into other co-ordinate
systems also. Its units are per square meter.
• It can be applied over vector as well scalar functions.
• When applied over a vector function, A it gives a vector whose Cartesian
components are the Laplacians of the Cartesian components of vectorA.
• When applied over a scalar function, it results in another scalar function.
By definition, Laplacian is, s2=s.s, called del squared

7. 2
2 2
2
o o
t
 

 

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Fields
Vector
Calculus-I
7
d'Alembertian
2 2 2
k k
     
• Operator d'Alembertian is indicated by a box/square  and,
sometimes by 2 . It is Laplace operator of Minkowski space and called
'box operator' and some times as 'quabla'.
• Helmholtz operator, named after Hermann von Helmholz is defined in
terms of Laplacian operator as,
• Both are applied over scalar functions.

8. Electromagnetic
Fields
Vector
Calculus-I
8
Del applied
Del can be applied over functions in three ways:
• Gradient: Del applied over a scalar function, giving a vector function
• Divergence: Del applied over a vector function, giving a scalar function
• Curl: Del applied over a vector function, giving a vector function

9. •Gradient results when Del is applied over a scalar function, giving a
vector.
Its magnitude is equal to the maximum change of function per
unit distance,
Its direction gives the direction of its maximum increase.
•If f is a scalar function, then its gradient is indicated as sf.
•According to a theorem on partial derivatives, differential df can be
expressed in terms of sf as,
l
df f d
  
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Gradient

10. •Divergence of an arbitrary vector field, A is defined as the outward
flux of A per unit volume, in the limit the volume shrinks to a point.
Mathematically,
•In this relation, closed surface integral denotes flux of A through a
closed surface s and τ is its volume.
• It's value can be found by applying del operator over, A through dot
operator , and is indicated as s A.
0
Div. of lim s
d
 
 



 A a
A
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Fields
Vector
Calculus-I
10
Divergence

11. •Curl of an arbitrary vector field, A is defined as the maximum
circulation of the field, A per unit area, in the limit of area shrinking to a
point. Mathematically,
Here, closed line integral in the numerator is circulation, Δa is its area
and is unit normal to the surface enclosed by the curve c.
•Curl direction is the direction of normal vector to the area in the limit
the area shrinks to a point and in the right hand sense.
•Its value can be found by applying del over A, through cross operator,
×, and is indicated as s×A.
max
a 0
ˆ
Curl of lim |
a
c
n
d
 



 A l
A n
n̂
Electromagnetic
Fields
Vector
Calculus-I
11
Curl

12. •Divergence is an indication of spreading/branching of field from a point
where as curl describes coiling/wrapping of field around a point.
•Divergence of a vector function gives a measure of its ability to cause
expansion/contraction where as the curl gives the ability to cause
turning/spinning.
•A field F is called solenoidal , if it is divergence-less i.e.s•F= 0 where
as it is called irrotational , if it is curl-less i.e. s×F=0.
Electromagnetic
Fields
Vector
Calculus-I
12
Div/Curl

13. ˆ ˆ ˆ
V V V
V
x y z
  
   
  
x y z
y
x z
D
D D
x y z

 
    
  
D
ˆ ˆ ˆ
x y z
x y z
D D D
  
 
  
x y z
D =
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Fields
Vector
Calculus-I
13
Del applied
( , , )
V x y z
Gradient: del applied over a scalar function
Divergence and Curl: del applied over a vector function
ˆ ˆ ˆ
( , , ) x y z
x y z D D D
  
D x y z
How to find them ????

14. S.No Name Solenoidal field Irrotational field
1 Feature Divergence-less Curl-less
2 Differential relation
3 Integral relation
4 Implication
5 Example fields Magnetic field Electrostatic field
0
F
  0
F
 
0
F a
d
 
 0
F l
d
 

F A
  F V
 
Electromagnetic
Fields
Vector
Calculus-III
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Field Nature

15. ENOUGH
FOR NOW
ENOUGH
FOR NOW
ENOUGH
FOR NOW
ENOUGH
FOR
NOW
ENOUGH
FOR
NOW
15