lecture6_gauss_law_and_fields_071811

lecture6_gauss_law_and_fields_071811 - Physics 142...

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Unformatted text preview: Physics 142 7/18/2011 The Electric Flux The electric flux measures the amount of electric field passing through a surface of area A whose normal to the surface is tilted at angle θ from the field. We can define the electric flux more concisely using the dot-product: The total electric flux through this box is 0% 0% 0% 0% 0% Nm2/C. B. 4 Nm2/C. C. 2 Nm2/C D. 1 Nm2/C. E. 0 Nm2/C. A. 6 Physics 142 Summer 2011 Dr. Nick Cummings 1 Physics 142 7/18/2011 Electric Flux rr so Φ e = E ⋅ A so units are N2 ⋅m C Area Area vector could point either way – Sign ambiguous so far If If the electric field lies in the plane of the surface flux is zero If E-field is perpendicular flux is just EA If E- Physics 142 Summer 2011 Electric Flux and Field Lines Geometrically: electric flux proportional to # of field lines passing through surface – Density of field lines proportional to field strength – (density of line)* area = # of lines Physics 142 Summer 2011 Dr. Nick Cummings 2 Physics 142 7/18/2011 Computing Flux r r Computing Computing flux is generally hard if E or A are changing at the surface – E-field changing direction – Surface curved BUT…. BUT…. Easy if everywhere on the surface the field is either perpendicular or parallel Physics 142 Summer 2011 Flux for a Closed Surface For For a closed surface define area vector to point out – Total flux is (flux out) – (flux in) Geometrically: Geometrically: (# of field lines out)-(# of out)field lines in) – Sources outside don’t contribute to flux – It doesn’t matter where the source is – Size and shape of surface don’t matter Physics 142 Summer 2011 Example: Point Source Flux Consider a point charge Q and a surface that is a sphere of radius R centered on that charge. What is the flux through that sphere? Physics 142 Summer 2011 Dr. Nick Cummings 3 Physics 142 7/18/2011 Gauss’s Law For any closed surface enclosing total charge Qin,the net electric flux through the surface is This result for the electric flux is known as Gauss’s Law. This box contains 0%A. 0%B. 0%C. 0%D. D. 0%E. E. a net positive charge. a net negative charge. a negative charge. a positive charge. no net charge. Physics 142 Summer 2011 These These are two-dimensional cross sections through threetwothreedimensional closed spheres and a cube. Rank order, from largest to smallest, the electric fluxes Φa to Φe through surfaces a to e. 0% A. 0% B. 0% C. 0% D. 0% E. Φa > Φc > Φb > Φd > Φe Φb = Φe > Φa = Φc = Φd Φe > Φd > Φb > Φc > Φa Φb > Φa > Φc > Φe > Φd Φd = Φe > Φc > Φa = Φb Physics 142 Summer 2011 Dr. Nick Cummings 4 Physics 142 7/18/2011 Who Cares About Flux? Gauss’ Gauss’ law relates charge and flux, but who cares about flux? – Generally flux integral is hard to compute – Often we want to know the E-field value at a pt E- If If the source is symmetrical, there may be a surface for which the E-field is always Eperpendicular or parallel – Calculating flux is easy – Can use flux to solve for E-field value EPhysics 142 Summer 2011 Using Flux to Find the Field Gauss’ Gauss’ law applies whether the surface is a physical object or just imaginary With With enough symmetry we can construct an imaginary surface where – Easy to calculate flux – Possible to solve for E-field value E- The The imaginary surface is called a “Gaussian surface” Physics 142 Summer 2011 Example: Charged Sphere Consider a sphere with charge Q distributed uniformly throughout it. Using Gauss’ law and symmetry, derive the formula for the electric field. Physics 142 Summer 2011 Dr. Nick Cummings 5 Physics 142 7/18/2011 Example: Charged Cylinder Consider an infinite cylinder with uniform charge per unit length λ distributed uniformly throughout it. Using Gauss’ law and symmetry, derive the formula for the electric field. Physics 142 Summer 2011 Which Gaussian surface would allow you to use Gauss’s law to determine the electric field outside a uniformly charged cube? A cube whose center coincides with the center of the charged cube and which has parallel faces. B. A sphere whose center coincides with the center of the charged cube. 0% C. Neither A nor B. D. Either A or B. A. A. B. C. D. Physics 142 Summer 2011 Dr. Nick Cummings 6 ...
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