Gauss's Law

• Electric Flux
• definition: number of field lines passing through a surface.
• number of lines will depend on
• strength of field
• area of surface
• orientation of surface with respect to field lines.
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• theta is the angle between the field lines and the direction perpendicular to the surface (shown below)



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• with this definition the area can be thought of as a vector. the electric field is a vector, definition is scalar product
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• this definition assumes a uniform electric field over the surface through which we want to determine the flux.
• If the electric field is not uniform
• break surface into little pieces, small enough so that the electric field is uniform over those pieces,
• calculate the flux through each of those pieces
• sum to find the total flux.



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• Gauss's Law
• flux through a closed spherical surface surrounding a charge, q
• let's put a point charge at the center of a spherical surface with radius R.





• we know the electric field a distance R away from a point charge.



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• this field is perpendicular to the surface of the sphere and also constant everywhere on the surface.
• to find the flux, we just multiply the electric field times the area of a sphere.



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• If we choose any other surface surrounding the charge, we must get the same flux.
• Why? Because the lines leave the positive charge and to get to infinity they must pass through the surface surrounding the charge. The number of lines leaving the charge does not depend on the shape or size of the surface surrounding the charge.
• In general Gauss's Law



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• This holds for any shape or any size surface. However Gauss's Law becomes very useful, when by symmetry, one can find a field over which the electric field is constant.



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• Then the problem becomes one of finding the area of a simple surface.



• Examples: Applying Gauss' Law for Cylindrical Symmetry, Planar Symmetry, and Spherical Symmetry