AP1200_Ch5_Electricity-2GaussLawPotentialEnergyCircuit-2008

# AP1200_Ch5_Electricity-2GaussLawPotentialEnergyCircuit-2008...

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Unformatted text preview: Electricity Electricity Lecture 2 Gauss’ Law 3 Flow Rate ( a ) A uniform air-stream of velocity v is perpendicular to the plane of a square loop of area A . ( b ) The component of v perpendicular to the plane of the loop is v cos θ , where θ is the angle between v and a normal to the plane. ( c ) The area vector A is perpendicular to the plane of the loop and makes an angle θ with v . ( d ) The velocity field intercepted by the area of the loop. ( 29 A v θ cos = Φ The Volume Flow Rate: A v ⋅ = Φ 4 Flux The word “flux” comes from the Latin word meaning “to flow.” Consider a uniform air- stream of velocity v, we can assign a velocity vector to each point in the air-stream passing through the loop. The composite of all those vectors is a velocity field , so we can interpret as giving the flux of the velocity field through the loop. With this interpretation, flux no longer means the actual flow of something through an area—rather it means the product of an area and the field across that area. A v ⋅ = Φ 5 Flux of an Electric Field A Gaussian surface of arbitrary shape immersed in an electric field. The surface is divided into small squares of area ∆ A . The electric field vectors E and the area vectors ∆ A for three representative squares, marked 1, 2, and 3, are shown. A E ∆ ⋅ = ∆ Φ ∫ ⋅ = Φ A E d The electric flux Φ through a Gaussian surface is proportional to the net number of electric field lines passing through that surface. 6 Question 5.7 The figure shows a Gaussian cube of face area A immersed in a uniform electric field that has the positive direction of the z axis. In terms of E and A , what is the flux through (a) the front face (which is in the xy plane), (b) the rear face, (c) the top face, and (d) the whole cube? 7 Water Source We represent the flow rate of water by lines. The closer the lines, the higher is the flow rate. We enclose the water source by a closed surface of arbitrary shape. The total number of lines enclosed by the surface is independent of the shape of the surface. It is only dependent on the strength of the source. 8 Gauss’ Law Gauss' law relates the net flux Φ of an electric field through a closed surface (a Gaussian surface) to the net charge q enc that is enclosed by that surface. The electric field can be pictured as field lines. The net charge is analogous to the water source in our previous example. ∫ = ⋅ enc q d A E ε Φ ∝ enc q Φ = ε enc q 9 Guassion Surface Surface S 1 The electric field is outward for all points on this surface. Thus, the flux of the electric field through this surface is positive, and so is the net charge within the surface....
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AP1200_Ch5_Electricity-2GaussLawPotentialEnergyCircuit-2008...

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