E & M Lecture 3 - Ch 24 Gausss Law Karl Friedrich...

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Karl Friedrich Gauss (1777-1855) – German mathematician Ch 24 – Gauss’s Law Ch 24 – Gauss’s Law
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Karl Friedrich Gauss (1777-1855) – German mathematician Already can calculate the E-field of an arbitrary charge distribution using Coulomb’s Law. Gauss’s Law allows the same thing, but much more easily… … so long as the charge distribution is highly symmetrical . Ch 24 – Gauss’s Law Ch 24 – Gauss’s Law
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Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux Electric Flux measures how much an electric field wants to “push through” or “flow through” some arbitrary surface area We care about flux because it makes certain calculations easier.
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Easiest case : The E-field is uniform The plane is perpendicular to the field E E A r Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 1 – Case 1 Electric Flux
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Easiest case : The E-field is uniform The plane is perpendicular to the field E E A r Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 1 – Case 1 Electric Flux Flux depends on how strong the E-field is and how big the area is.
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Junior Varsity case : The field is uniform The plane is not perpendicular to the field E E A E Acos r r Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 2 – Case 2
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Junior Varsity case : The field is uniform The plane is not perpendicular to the field E E A E Acos r r Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 2 – Case 2 Flux depends on how strong the E-field is, how big the area is, and the orientation of the area with respect to the field’s direction.
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E E A E Acos r r ˆ n ˆ A An r A E E And, we can write this better using the definition of the “dot” product. where: n A A ˆ Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 2 – Case 2
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A E E Quick Quiz: What would happen to the E-flux if we change the orientation of the plane? Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 2 – Case 2
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E r E r A A Varsity (most general) case : The E-field is not uniform The surface is curvy and is not perpendicular to the field Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 3 – Case 3
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E r E r A A Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 3 – Case 3 Imagine the surface A is a mosaic of little tiny surfaces ΔA . Pretend that each little ΔA is so small that it is essentially flat.
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E r E r A A Ch 24.1 – Electric Flux Ch 24.1 – Electric Flux – Case 3 – Case 3 Then, the flux through each little ΔA is just: A E E is a special vector. It points in the normal direction and has a magnitude that tells us the area of ΔA .
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