Phys2212_L07 - Physics 122B Electricity and Magnetism...

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Physics 122B Electricity and Magnetism Lecture 7 Gauss’s Law and Applications
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10/06/09 Physics 2212 - Lecture 7 2 Flux and Charge Consider the flux through a spherical surface of radius R that surrounds a point charge q located at its center. This result is independent of how big we choose to make the spherical surface. A very large or a very small surface will have the same flux, which depends only on the size of the charge q at the center of the sphere. Essentially, q/ ε 0 is the number of field lines attached to charge q , and ε 0 is the charge producing one unit of flux. ( 29 sphere 2 2 0 0 1 4 4 e E dA E A q q R R π πε ε Φ = = = = r r r r
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10/06/09 Physics 2212 - Lecture 7 3 Flux is Independent of Surface Shape and Radius What about the flux when the surface has an arbitrary non-spherical shape? We can approximate such a surface by a set of spherical pieces. Summing over these gives Φ e = q / ε 0 . We conclude that the flux through an arbitrary surface always depends only on the charge q enclosed by the surface.
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10/06/09 Physics 2212 - Lecture 7 4 Charge Outside the Surface What about the flux through a surface from any charges not enclosed by the surface? There is no net flux because entering and exiting fluxes cancel.
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10/06/09 Physics 2212 - Lecture 7 5 Multiple Charges So far we have considered only the flux from a single point charge. What about the case of multiple charges? in 1 2 3 Q q q q + + + L 1 2 3 1 2 3 3 1 2 0 0 0 e E dA E dA E dA E dA q q q ε Φ = = + + + = Φ + Φ + Φ + = + + + r r r r r r r r L L L in 0 Therefore, e Q E dA Φ = = r r
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10/06/09 Physics 2212 - Lecture 7 6 Gauss’s Law The previous arguments lead to the conclusion that the correspondence between the charge Q in enclosed by a surface and the net flux Φ e through that surface is a general result. It is called Gauss’s Law , and is usually written as: Gauss’s Law is the first of four master equations, collectively called Maxwell’s Equations , that together constitute a “unified field theory” of electromagnetism. In essence, Gauss’s Law says that diverging field lines from a point indicate the presence of an electric charge at that point, and that this charge can be “detected” by surrounding the point with a surface and observing the flux through the surface. Johann Carl Friedrich Gauss (1777 – 1855) in 0 e Q E dA ε Φ = = r r
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10/06/09 Physics 2212 - Lecture 7 7 Gaussian Spherical Geometry This means that the electric field can have only radial components and can depend only on radius. If a concentric spherical Gaussian surface is constructed around the charge distribution, the E -field and dA surface vectors will be parallel, and the magnitude of the E -field will be constant over the surface. Therefore, Φ e = E·dA = EA , greatly simplifying the flux calculation. Spherical distributions of charge have spherical symmetry, so the electric field must also have spherical symmetry.
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10/06/09 Physics 2212 - Lecture 7 8 Example : E Inside a Sphere of Charge Q R r x What is the electric field E i at some point x inside a hollow
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Phys2212_L07 - Physics 122B Electricity and Magnetism...

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