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PHY2049ch23A%281-20-10%29%20corrected

# PHY2049ch23A%281-20-10%29%20corrected - Gauss Law of...

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Gauss’ Law of Electrostatics Johann Carl Friedrich Gauss 1777 – 1855 A more advanced method to evaluate the electrostatic field in the region surrounding an assembly of charge. Child prodigy, Mathematician, Physicist Among his great accomplishments was a key realization , that with his mathematical abilities he was able to formulate into a central law of electrostatics . enc o surface q EdA ⋅= ε G G v

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The realization was the following: Suppose that some charge exists in a region of space. Now consider a closed surface that encloses a volume of arbitrary shape. If the volume contains no net charge inside it, the number of field lines that enter the closed surface has to equal the number of field lines that exit the closed surface. + If we assign the field lines that enter the closed surface a negative sign , and those that exit a positive sign then the sum of all the field lines over the entire surface is zero .
If, however, the closed surface encloses net charge , then the number of field lines that exit (+q enc ) or enter (-q enc ) the surface is proportional to the quantity of charge enclosed. +

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Hence (following Gauss) if we quantify , and sum the electric field that crosses the closed surface then we can tell the quantity of charge enclosed. Alternatively, if the quantity and distribution of charge is known , and the closed surface (which recall can have arbitrary shape) takes on a symmetry appropriate to the charge distribution, then (as we’ll see) the electric field can be determined . To do this we must develop the concept of electric flux . The electric flux, Φ , is the signed quantity of total electric field that penetrates a surface of area Δ A.
The flux will depend on: the magnitude of the electric field , E , the size of the surface (area, Δ A ) and the angle , θ between the electric field vector and the surface. To satisfy such dependence the electric flux is defined as, ˆ EA E n A E ( A ) c o s Φ= ⋅Δ = ⋅ Δ = Δ θ G G G Where ˆ n ˆ n ˆ n A Δ A Δ A Δ ˆ AA n Δ= Δ G has direction of the surface normal . ˆ n E G E G E G

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There is an ambiguity here in that the surface normal can be in either of two directions (both perpendicular to the surface). This ambiguity is removed once we consider a closed surface (called a Gaussian surface ) for which the surface normal is taken to have direction pointing out of the closed surface volume. If the surface is curved it can be divided into approximately rectangular grid elements where each element is flat and the curvature is accommodated by small angles between the adjacent flat elements.
The flux of the electric field through this surface is then approximately the sum of all the flux calculated for each element, i.e., EA Φ≈ ⋅Δ G G This approximation becomes exact if we let the grid spacing shrink to make the area elements infinitesimal .

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PHY2049ch23A%281-20-10%29%20corrected - Gauss Law of...

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