Chapter_28_lect_pls - Rayleigh criterion Chapter 28 Gausss...

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Chapter 28 Gauss’s Law Rayleigh criterion
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Gauss’s Law and Electric Flux The electric flux through any closed surface is proportional to the charge enclosed. Charge outside the closed surface contributes zero net flux. 0 in e q d ε Φ= = EA electric flux is a scalar quantity: Φ e units: N m 2 C −1 What is electric flux? analogous to the flux in moving fluids (see next slide) Mathematical statement of Gauss’s Law In practice we will only use the integral when symmetry renders it is easy to do so. Gauss’s Law is useful for analysis of some basic electrostatic problems. Although it is mathematical, we will apply it in an almost conceptual manner.
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Flux Definition The amount of stuff crossing a given area per unit time. If the flow (velocity vector v ) is perpendicular to the area, the maximum amount of fluid flows through the loop. The volume of fluid per unit time (flux) is then Φ = vA . (units: m s −1 m 2 = m 3 s −1 ) We can also right this as a vector equation: Consider the flow of a fluid. The flux is the volume of fluid crossing the area per unit time. Imagine an area defined by the brown loop with area A . Area is a vector A perpendicular to the surface. We also define the normal unit vector , such that . ˆ n ˆ An = A ( ) ˆ cos cos 0 A n vA vA vA θ Φ= = = = vA v  If the flow (velocity vector v ) is parallel to the area, no fluid flows through the loop. The volume of fluid per unit time (flux) is then Φ = 0 . We can also right this as a vector equation: ( ) ˆ cos cos 90 0 A n vA vA = = = v flow area Ө = 0 Ө = 90 Remember the dot product returns a scalar
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Flux Definition The amount of stuff crossing a given area per unit time. The general case. If the flow (velocity vector v ) makes an angle to the surface normal, the flux through the loop is reduced by cos Ө of that angle. You can visualize this in two equivalent ways: From the perspective of the flow, a loop whose area is at an angle to the flow intercepts a smaller area of the flow. The projection of the area onto the flow is A = A cos Ө , thus the Φ = vA = v ( A cos Ө ) . From the perspective of the loop, a flow passing through the loop at an angle has an effectively lower velocity through the loop. The projection of the velocity onto the surface normal is v = v cos Ө , thus the Φ = v A = ( v cos Ө ) A . Both of these views is mathematically equivalent and can be succinctly stated as the dot product of v and A . Consider the flow of a fluid. The flux is the volume of fluid crossing the area per unit time. Imagine an area defined by the brown loop with area A . Area is a vector A perpendicular to the surface. We also define the normal unit vector , such that . ˆ n ˆ An = A ˆ cos A n vA θ Φ= = = vA v  Only the component of v parallel to the surface normal contributes to the flux.
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Electric Flux uniform E and a plane surface Electric field is analogous to flow velocity, although the electric field is not actually flowing.
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Chapter_28_lect_pls - Rayleigh criterion Chapter 28 Gausss...

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