Lec 02 2015 electromagnetic : Chapter 1 (cont.)

1. Electromagnetic Field Theory
2nd Year EE Students
Prof. Dr. Magdi El-Saadawi
www.saadawi1.net
saadawi1@gmail.com
2014/2015

2. Chapter 1
VECTOR ALGEBRA
(Continue)
10/16/2014
Prof. Dr. Magdi El-Saadawi
2

3. 1.5. Vector Multiplication
Vectors may be multiplied by scalars: The magnitude of the vector changes, but its direction does not when the scalar is positive.
In case of vector multiplication:
the dot product (also called scalar product)
the cross product (also called vector product).
10/16/2014
Prof. Dr. Magdi El-Saadawi
3

4. 1.5.1 The dot Product
Two vectors and are said to be orthogonal (or perpendicular) with each other if 10/16/2014
Prof. Dr. Magdi El-Saadawi
4

5. 1.5.1 The dot Product
The dot product obeys the following identities: 10/16/2014Prof. Dr. Magdi El-Saadawi5

6. 1.5.1 The dot Product
The most common applicationof the dot product is:
The mechanical work W, where a constant force F applied over a straight displacement L does an amount of work i.e.
Another example is the magnetic fields Φ, where
10/16/2014
Prof. Dr. Magdi El-Saadawi
6

7. 1.5.2 The cross product10/16/2014
Prof. Dr. Magdi El-Saadawi
7

8. 1.5.2 The cross product10/16/2014Prof. Dr. Magdi El-Saadawi8

9. 1.5.2 The cross product
10/16/2014
Prof. Dr. Magdi El-Saadawi9

10. 1.5.2 The cross product10/16/2014
Prof. Dr. Magdi El-Saadawi
10

11. 1.5.2 The cross product10/16/2014
Prof. Dr. Magdi El-Saadawi
11

12. 1.5.2 The cross product10/16/2014
Prof. Dr. Magdi El-Saadawi
12

13. 1.6. The Gradient
The gradient of a scalar field is:
a vector field that lies in the direction for which the scalar field is changing most rapidly. The magnitude of the gradient is the greatest rate of change of the scalar field.(see figure 1.9 pp. 19) 10/16/2014Prof. Dr. Magdi El-Saadawi13

14. 10/16/2014
Prof. Dr. Magdi El-Saadawi
14

15. 1.6. The Gradient10/16/2014
Prof. Dr. Magdi El-Saadawi
15

16. 1.6. The Gradient10/16/2014Prof. Dr. Magdi El-Saadawi16

17. 1.6. The Gradient10/16/2014Prof. Dr. Magdi El-Saadawi17

18. 10/16/2014Prof. Dr. Magdi El-Saadawi
18

19. 1.7. Divergence of a vector and Divergence Theorem
The flux
Assume a vector field A, continuous in a region containing the smooth surface S, we define the surface integral of the flux of through S as:
Or
For a closed surface
10/16/2014
Prof. Dr. Magdi El-Saadawi19

20. 10/16/2014Prof. Dr. Magdi El-Saadawi20

21. 1.7. Divergence of a vector and Divergence Theorem10/16/2014Prof. Dr. Magdi El-Saadawi21

22. If there are no sources within the boundary surface, thus the integral will get the value zero (Fig. b)
If there is a source (or sink) within the surface of integration, which generates new field lines the integral will get a value different from zero (Fig. a).
10/16/2014
Prof. Dr. Magdi El-Saadawi
22

23. 10/16/2014Prof. Dr. Magdi El-Saadawi23

24. 1.7. Divergence of a vector and Divergence Theorem
The theorem of Gauss(Divergence theorem) is proved from the definition of the divergence and it enables to transform surface integrals into volume integralsas follows:
The volume integral about a specific flux from an element of volume V is equal to the flux thorough going from the closed surface S bounding this volume (V).10/16/2014Prof. Dr. Magdi El-Saadawi24

25. Examples
10/16/2014
Prof. Dr. Magdi El-Saadawi
25

26. 10/16/2014Prof. Dr. Magdi El-Saadawi26

27. Examples
10/16/2014
Prof. Dr. Magdi El-Saadawi27

28. Examples10/16/2014Prof. Dr. Magdi El-Saadawi
28

29. Examples
10/16/2014
Prof. Dr. Magdi El-Saadawi29