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L07_Viet_Gauss&Acirc;&brvbar;s Law and Applications

# L07_Viet_Gauss&Acirc;&brvbar;s Law and Applications...

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Physics 122 Electricity and Magnetism Lecture 7 Gauss’s Law and Applications

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05/12/09 Physics 122 - 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
05/12/09 Physics 122 - 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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05/12/09 Physics 122 - 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.
05/12/09 Physics 122 - 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 E + + + 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 ε ε Φ = = + + + = Φ + Φ + Φ + = + + + r r r r r r r r L L L in 0 Therefore, e Q E dA ε Φ = = ° r r

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05/12/09 Physics 122 - 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
05/12/09 Physics 122 - Lecture 7 7 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 spherical shell of radius R with total charge Q distributed uniformly over the shell? Put a concentric spherical Gaussian surface through x , so that it has radius r<R . From the spherical symmetry of the charge distribution, the electric field E i on this surface must be strictly radial (in spherical coordinates).

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