anonymous 3 years ago Partial Differential Problem

1. anonymous

2. anonymous

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 $$\rho$$.

3. anonymous

i typed, have idea, @rox13kh ???

4. anonymous

hei where are you going @rox13kh ??

5. anonymous

@UnkleRhaukus @oldrin.bataku have idea ??

6. anonymous

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}}$

7. anonymous

in a similar process$\nabla . (- \epsilon \nabla \phi)=\rho$ in a region of charge density $$\rho$$ . if $$\epsilon$$ is constant$-\epsilon \ \nabla^2 \phi=\rho \ \ \ \ \color\Green{\text{ Poisson's Equation}}$

8. anonymous

ah thank you @mukushla ..., i have another one .., wanna help me again?