A pictorial method of visualizing curl & determinant operation utilized in electromagnetics
- 1. International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol.2, No.1, February 2013 21 A pictorial method of Visualizing Curl & Determinant operation utilized in Electromagnetics 1 Raveesha KH and 2 Dr V H Doddamani 1 Faculty, Dept. of Physics CMR Institute of Technology, Bangalore hod.physics@cmrit.ac.in 2 Associate Professor, Dept. of Physics, Bangalore University, Bangalore drvhdmani@gmail.com ABSTRACT Curl and Determinant operations are well known in the field of Electromagnetics and fluid dynamics. Famous Maxwell’s equations involve curl operation. Though this operation is widely used to represent rotation, we do not have sufficient literature explaining how this operation represents rotation. Most of the books omit detailed discussion on the physical interpretation of the Curl and Determinant operation. In this paper we have attempted to develop a pictorial method to provide a logical proof showing the versatility of this operation to study the rotating vectors. We further show how this pictorial method simplifies the fundamental expression with regard to determinants. KEY WORDS Vector, Curl, Determinant, clockwise, anticlockwise, Introduction 1.Introduction The curl and Determinant tools are widely used in Mathematics and Physics. Through the author’s interaction with engineering students over a period of many years, it is observed that students are acquiring the skill of solving problems involving Curl and Determinants rather than appreciating the concept. This prompted the authors to propose a pictorial method to visualize the physical significance of these mathematical tools. The pictorial method as to how the rotation of a vector is studied through Curl operation is described. In the second section the method is described with the help of diagrams. Explanation on the fundamental expression used to evaluate a determinant is provided. Conclusions are mentioned at the end. 2. Method We know that Curl of a vector B is represented using determinants in the following way
- 2. International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol.2, No.1, February 2013 22 zyx BBB zyx zˆyˆxˆ B ∂ ∂ ∂ ∂ ∂ ∂ =×∇ This determinant is evaluated using the formula ∂ ∂ − ∂ ∂ + ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ =×∇ y B x B z z B x B y z B y B xB xyxzyz ˆˆˆ …… (1) Here we need to explore as to how this expression represents the rotation of the vector B. Consider a vector whose direction changes in space as shown in the figure 1 given below Fig 1 It is proposed that Curl of this vector B is written as Net Rotation Net Rotation Net Rotation Here, it can be pictorially shown that each term on the right hand side represents net rotation of a component of the vector B. Let us consider the first term ∂ ∂ − ∂ ∂ + ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ =×∇ y B x B z z B x B y z B y B xB xyxzyz ˆˆˆ
- 3. International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol.2, No.1, February 2013 23 ∂ ∂ − ∂ ∂ z B y B yz Anticlockwise Rotation Clockwise Rotation The term shows the variation of Z component of the vector B along Y axis. In the following figure 2, the variation of the vector B along Y axis from P to Q is shown for a case where in the Z component of the Vector B is increasing along Y axis. From this figure, it is clear that the term is the measure of magnitude of rotation of vector B in anticlockwise direction in the spatial range P to Q along Y axis. Fig 2 Now consider the term . .This term shows the variation of Y component of the vector B along Z axis. In the following figure 3, the variation of vector B along Z axis from P to Q for a case where in Y component of the Vector B is increasing along Z axis. From this figure, it is clear that the term is a measure of degree of rotation of the vector B in clockwise direction in the spatial range P to Q along Z axis. Fig 3 Similarly, the second and third terms in the expression (1) may also be shown to be representing clockwise/anticlockwise rotation. From this discussion, we can now write determinant expression as y Bz ∂ ∂ y B z ∂ ∂ ∂ ∂ z By ∂ ∂ z By
- 4. International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol.2, No.1, February 2013 24 ∂ ∂ − ∂ ∂ + ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ =×∇ y B x B zˆ z B x B yˆ z B y B xˆB xyxzyz Anticlockwise Rotation –Clockwise Rotation Clockwise Rotation – Anticlockwise Rotation Anticlockwise Rotation – Clockwise Rotation Net Rotation Net Rotation Net Rotation This discussion can also be extended to logically understand the negative sign present in the second term. Here, it is to be noticed, that, in the case of second term, the symmetry is violated i.e., in the first and third terms, anticlockwise rotation is considered first, where as in the second term, clockwise rotation is considered first. 3. Results and Discussion Curl is a fundamental operation utilized in Electromagnetics to represent rotation. Through the diagrams described in this paper, the rotation of magnetic field vector and in general any force field is described. This pictorial method makes use of the fact that derivative of a component of a function along a perpendicular axis represents the change in the magnitude of the vector function. This change is attributed to the rotation of the vector function. Thus curl becomes an useful tool to represent rotation. 4. Conclusions Curl and the determinants may be utilized to represent changing directions /rotation of a vector. A pictorial scheme of describing rotation of a vector using the mathematical tools like CURL and Determinants is presented. This pictorial scheme suggested enables students to visualize the physical significance of mathematical tools like CURL and Determinants. Bibliography 1. Michael Faraday, Phil. Trans. R. Soc. London. 1832, 122, 125-162 2. Oliver Heaviside, Electromagnetic Theory, London, Chelsea,1897 3. David J. Griffiths’, Introduction Electrodynamics (3rd Edition), New Delhi, Prentice Hall of India, 2004 4. D. Cheng, Field and wave Electromagnetics ( 2nd Edition), New Delhi, Pearson Education (India),2002 5. Nathan Ida, Engineering Electromagnetics( 2nd Edition), New Delhi, Springer (India) ,1998 6. D. Chattopadyay et al., Electricity and Magnetism, Calcutta, Central, 2006, 7. John D.Kraus, Electromagnetics ( 3rd Edition), New York, McGraw-Hill, 2001 8. J .D. Jackson, Classical Electrodynamics (3rd Edition), New Delhi, Wiley India Edition,2000
- 5. International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol.2, No.1, February 2013 25 9. Feynman, Lectures on Physics, New Delhi, Narosa Publishing House,1999 10. Guru & Hiziroglu, Electromagnetic Field Theory Fundamentals, New Delhi, Vikas publishing House, 2003 11. Erik Hallen, Electromagnetic Theory, London, Chapman & Hall,1962 Authors: Raveesha KH Faculty Department of Physics CMR Institute of Technology IT Park Road Kundalahalli Bangalore-37 Karnataka state India Raveesha KH has been teaching Physics and Electromagnetics courses for engineering students from the past 16 years. I am currently involved with designing lab experiments for undergraduate students. I am pursuing research in the field of Radio astronomy at Bangalore University and Indian Institute of Astrophysics, Bangalore. My interests include designing astronomy related experiments. Dr Vijayakumar H Doddamani Associate Professor Department of Physics Bangalore University Jnanabharathi campus Off Mysore Road Bangalore-56 Karnataka state India Dr V H Doddamani has been teaching post graduate and doctoral students from the past 20 years. My research areas include Active Galactic Nuclei, Radio astronomy, Plasma Physics etc. Currently, I am involved in designing Radio astronomy related experiments.