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Partial Differential Problem

Mathematics
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Assume from electricity the equations \(\nabla . \bar D = \rho \) (\(\bar D \) = electric displacement ) and \(\rho \) = charge density and \(\bar D = - \epsilon \nabla \phi \), \(\phi \) = electrostatic potential and \(\epsilon\) = dielectric constant. Show that the electric potential satisfies laplace's equations in a charge -free region and satisfies poisson's equation in a region of charge density \(\rho\).
i typed, have idea, @rox13kh ???

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Other answers:

hei where are you going @rox13kh ??
@UnkleRhaukus @oldrin.bataku have idea ??
in a charge-free region \(\rho=0\) and u have\[\nabla . \bar D = \rho=0\]and we know \(\bar D = - \epsilon \nabla \phi\) so\[\nabla . (- \epsilon \nabla \phi)=0\]\[\nabla . ( \nabla \phi)=0\]\[\nabla^2 \phi=0 \ \ \ \ \color\red{\text{Laplace Equation}}\]
in a similar process\[\nabla . (- \epsilon \nabla \phi)=\rho\] in a region of charge density \(\rho\) . if \(\epsilon\) is constant\[-\epsilon \ \nabla^2 \phi=\rho \ \ \ \ \color\Green{\text{ Poisson's Equation}}\]
ah thank you @mukushla ..., i have another one .., wanna help me again?

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