Derivation of Poisson's Equation and Laplace's Equation

[MadMike1986]

Hi,

Can someone point me in the right direction to a derivation of Poisson's Equation and of Laplace's Equation, (from Maxwell's equations I think) both in a vacuum and in material media?

How does one get from Maxwell's equations to Poisson's and Laplace's?

 

[Nabeshin]

So the two relevant maxwell's equations are:
$$\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon}$$
$$\vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$$

For an electrostatic system, there is no changing B field so,
$$\vec{\nabla} \times \vec{E} = 0$$
Which implies E can be written as the gradient of a scalar potential,
$$\vec{E} = - \vec{\nabla} \varphi$$

Combining this fact with the first equation,
$$\vec{\nabla} \cdot \vec{\nabla} \varphi = \nabla^2 \varphi = - \frac{\rho}{\epsilon}$$

And of course Laplace's equation is the special case where rho is zero.

Cheers!

 

[MadMike1986]

Ah, thank you very much. That's not so bad after all.