7B_Week_6_Gauss

7B_Week_6_Gauss - E&M Week 2: Outline Contiguous Charge...

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E&M Week 2: Outline Contiguous Charge distributions Gauss’s Law Electrical Potential In part thanks to MIT Technology Enabled Active Learning
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Review: Continuous Charge Distributions Break distribution into parts: Superposition: E = Δ E d E Q = Δ q i i dq V ∫∫∫
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R: Continuous Sources: Charge Density
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Line of Charge Same physics, nasty trig substitution…
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E Field from Line of Charge E = k e Q s ( s 2 + L 2 / 4) 1/ 2 ˆ j Limits: s >> L lim E k e Q s 2 ˆ j s << L lim E 2 k e Q Ls ˆ j = 2 k e λ s ˆ j Point charge Infinite charged line
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Uniformly Charged Disk P on axis of disk of charge, x from center Radius R , charge density σ . Find E at P
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Disk: Two Important Limits E disk = σ 2 ε o 1 x x 2 + R 2 ( ) 1/ 2 ˆ i x >> R lim E disk 1 4 πε o Q x 2 ˆ i x << R lim E disk 2 o ˆ i Limits: Point charge Infinite charged plane
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E for Plane is Constant? 1) Dipole: E falls off like 1/r 3 2) Point charge: E falls off like 1/r 2 3) Line of charge: E falls off like 1/r 4) Plane of charge: E constant
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Now we can… Treat 1 to Infinite number of charges. Most processes in nature involve lots of charges Lightning, neural computation, etc.
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Brief comment on neurons Voltage gated ion channels:
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Quick Review Three things still needed: 1. Continuous charge distributions (done) 2. Gauss’ law 3. Conservation of energy (potentials, work, etc.)
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Gauss’s Law The first Maxwell Equation An essential computational technique This is important!
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Gauss’s Law – The Idea The total “flux” of field lines penetrating any of these surfaces is the same and depends only on the amount of charge inside
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Flux Water through a faucet
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Electric Flux Φ E Case I: E is constant vector field perpendicular to planar surface S of area A
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Case II: E is constant vector field directed at angle θ to planar surface S of area A Electric Flux Φ E
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Gauss’s Law – The Equation Electric flux Φ E (the surface integral of E over closed surface S) is proportional to charge inside the volume enclosed by S
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