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METHOD OF IMAGES

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Class Activities: Method of Images

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Whiteboard: Calculate voltage (everywhere in space!) for 2 equal/opposite point charges a distance “d” above and below the origin. Where is V(r)=0?

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A) Simple Coulomb’s law: B) Something more complicated A point charge +Q sits above a very large grounded conducting slab. What is E(r) for other points above the slab? z x y +Q 3.7 r

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Infinite grounded conducting slab

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Boundary Conditions: (grounded)

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Calculate voltage V(r) (everywhere in space!) for 2 equal/opposite point charges a distance “d” above and below the origin. Where is V(r)=0? (If you’re done early, figure out Ex, Ey, and Ez from this voltage, and evaluate (simplify) AT the plane z=0)

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Boundary Conditions: (grounded)

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Boundary Conditions: “Image Charge” (grounded)

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Method of Images Last Class: Uniqueness Theorem If: & & If: on the boundaries Poisson’s Equation Then:everywhere (within the boundaries)

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Boundary Conditions: Uniqueness Theorem (grounded)

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Siméon Denis Poisson 1781-1840 Method of Images

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William Thomson (Lord Kelvin) 1824-1907 Method of Images

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James Clerk Maxwell 1831-1879 “Treatise on Electricity and Magnetism” 1873, Vol. 1, Ch. XI (p. 245-283, 3 rd Ed.) Method of Images

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Boundary Conditions: (grounded)

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Boundary Conditions: (grounded)

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A) 0 B) C) D) Something more complicated A point charge +Q sits above a very large grounded conducting slab. What is the electric force on this charge? z x y +Q d 3.8

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A) B) Something else. A point charge +Q sits above a very large grounded conducting slab. What's the energy of this system? +Q 3.8b z x y d

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3.9 Two ∞ grounded conducting slabs meet at right angles. How many image charges are needed to solve for V(r)? +Q A) one B) two C) three D) more than three E) Method of images won't work here r

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On paper (don’t forget your name!) in your own words (by yourself): What is the idea behind the method of images? What does it accomplish? What is its relation to the uniqueness theorem?

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Could we use the method of images for THIS problem? Find V(r) everywhere z>0, Given these two charges above a (grounded, infinite, conducting) plane? A)Yes, it requires 1 “image charge” B)Yes, it requires 2 images C)Yes, more than 2 image charge D) No, this problem can NOT be solved using the “trick” of image charges…

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Could we use the method of images for THIS problem? Find V(r) everywhere z>0, Given these two charges above a (grounded, infinite, conducting) plane?

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