FWLec3 - A. F. Peterson: Notes on Electromagnetic Fields &...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 9/04 Fields & Waves Note #3 Vector Integrals Objectives: Introduce vector line integrals and discuss the process of obtaining the differential length vector for arbitrary curves. Introduce vector surface integrals and the differential surface area vector. Electromagnetic fields are described by Maxwell’s equations. The integral form of these equations involve line and surface integrals. In preparation for our later study of Maxwell’s equations in integral form, this Note considers line and surface integrals. Line Integrals In physics, students learn that work is the product of force times distance. Force is actually a vector quantity, and the relation can be better expressed as WF d = (3.1) where W is the work, F is the force vector, and d is the displacement vector along a straight-line path. If the work is done along a curved path, the path can be approximated by N short segments as shown in Figure 1, and the work can be approximated from the summation d n n N @ =  1 (3.2) In the limiting case as the segment lengths decrease to better approximate the curve, equation (3.2) becomes the integral d path = Ú l (3.3) d l is a differential length vector. This quantity has units of length, and is a vector that is tangential to the curve at each point. The differential length vector The general expression for the differential length vector is given in the Cartesian coordinate system by dd x x d y yd z z l =++ ˆˆˆ (3.4)
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A. F. Peterson: Notes on Electromagnetic Fields & Waves 9/04 For most calculations, equation (3.4) must be specialized to the particular path being considered. The following examples illustrate the procedure. Example: Find the differential length vector d l for the curve yx z == , 0 (3.5) Solution: The curve in equation (3.5) is a parabola opening around the x -axis, as plotted in Figure 2. From the equation for the curve, we differentiate to obtain dy x dx dz - 1 2 0 12 / (3.6) Equation (3.6) provides a constraint that can be used to specialize equation (3.4) to the particular curve in question. By substitution, we obtain dd x x d y yd z z dx x x dx y z x x x l =++ =+ Ê Ë Á ˆ ¯ ˜ +() Ê Ë Á ˆ ¯ ˜ - ˆˆˆ ˆˆ ˆ / 1 2 0 1 2 (3.7) We have now expressed d l entirely in terms of dx , which is appropriate if we intend to use x as the integration variable. As an alternative, we could leave d l entirely in terms of dy . From equation (3.6), we obtain the equivalent relation dx x dy y dy dz = 22 0 , (3.8) By substitution into (3.4), we obtain x x d y z z ydy x dyy z yx y dy l = () ++ ˆ 20 2 (3.9) We have now expressed d l entirely in terms of dy , which is appropriate if we intend to use y as the integration variable.
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A. F. Peterson: Notes on Electromagnetic Fields & Waves 9/04 It is worthwhile to verify that the d l vectors we obtain are in fact tangent vectors to the curve in (3.6). By evaluating the preceding expressions at specific points along the curve, we should be able to verify their directions.
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This note was uploaded on 01/27/2011 for the course ECE 3025 taught by Professor Citrin during the Spring '08 term at Georgia Institute of Technology.

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FWLec3 - A. F. Peterson: Notes on Electromagnetic Fields &...

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