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Demo03Images - ECE 3030 Electromagnetic Fields and Waves...

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 3 INSTRUCTOR NOTES: Images and Poisson’s Equation Reminder: Images will be used here and later! Date: Tuesday 9/15 1. Call attention to trig identities in appendix of text (Put one on board?): sin θ = + cos( θ - π 2 ) = - sin( θ - π ) = - cos( θ - 3 π 2 ) cos θ = - sin( θ - π 2 ) = - cos( θ - π ) = + sin( θ - 3 π 2 ) cos( 3 π 4 - θ ) = cos( π 4 + θ 0 ) = sin( θ - π 4 ) = - cos( θ + π 4 ) where θ 0 = π 2 - θ (complement) sin( α + β ) sin( α - β ) = sin 2 α - sin 2 β = cos 2 β - cos 2 α cos( α + β ) cos( α - β ) = cos 2 α - sin 2 β = cos 2 β - sin 2 α 2. Taylor’s Series Expansions (Students should already know about these, but usually need this reminder!) Useful for quick approximations. f ( x ) = f (0) + 1 1! f 0 (0) x + 1 2! f 00 (0) x 2 + 1 3! f 000 (0) x 4 + ... Example: f 1 ( x ) = 1 1 - x f (0) = 1 f 0 1 ( x ) = - 1 (1 - x ) 2 ( - 1) = 1 (1 - x ) 2 f 0 1 (0) = 1 f 00 1 ( x ) = - 2 1 (1 - x ) 3 ( - 1) = 2 (1 - x ) 3 f 00 1 (0) = 2 f 1 ( x ) = 1 1 - x 1 + x + x 2 ... -→ f 2 ( x ) = 1 1 + x 1 - x + x 2 ... 3. Distances – reminders: Draw fig and ask what is d ? d = | ~ r 1 - ~ r 2 | d = £ ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 + ( z 1 - z 2 ) 2 / 1 / 2 CORNELL UNIVERSITY c WES SWARTZ (09/8/11; R09/9/5) 3–1
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ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 3 INSTRUCTOR NOTES: Images and Poisson’s Equation 4. Problem 4.3: Consider a point charge placed near the corner of perfect metal planes, as show in the figure below. The metal planes are infinite in the y direction. The point charge is placed at y = 0. a. Using the method of images draw the image charges corresponding to the point charge and indicate the lo- cation, orientation and magnitude of these image charges in a sketch. perfect conductor 2 d 2 d x z θ q + r & Figure 1: A charge on the bisector of two perpendicular conducting planes. b. Find the potential Φ( ~ r ) outside the perfect metal at a location ~ r far away from the charge (where | ~ r | > d ) in the x - z plane (i.e. for the angle φ in the spherical co-ordinates equal to zero). Use spherical coordinates.
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