# div_4 - ∇ − = and = ⋅ ∇ E so that = ∇ − ⋅ ∇...

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PHY2061 R. D. Field Department of Physics div_4.doc Univesity of Florida Gauss’ Law (integral form) The total electric flux, Φ E , through any closed surface S is equal to the total charge enclosed by the surface S (divided by ε 0 ) as follows: = = Φ S enclosed E Q dA E 0 ε Proof: 0 0 1 ) ( ρ enclosed V V S E Q dV dV E A d E = = = = Φ ! ! ! ! , where the total enclosed charge is given by = = V V enclosed dV dQ Q . Poisson’s Equation The electrostatic field can be written as the gradient of the electric potential, V(x,y,z), as follows: V E
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Unformatted text preview: ∇ − = ! ! and = ⋅ ∇ E ! ! so that ) ( = ∇ − ⋅ ∇ V ! ! or 2 2 2 2 2 2 2 − = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ z V y V x V V . Laplace’s Equation Whenever = , that is, in all parts of space containing no electric charge the electric potential must satisfy, 2 2 2 2 2 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ z V y V x V V . Any Closed Surface S E Gauss’ Law Laplace’s Equation Poisson’s Equation...
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