# When do I use Poisson's equation and when do I use Laplace's equation?

If I have a charged body near a conducting cylinder, am I supposed to used Poisson's or Laplace's equation? I know that I am supposed to use Laplace's equation only when the total charge density is zero, but I was thinking if I have the description of the electric field, I could do it just like the case of a sphere in a constant electric field where we can use Laplace's equation. I don't understand when to use Poisson's equation rather than Laplace's.

• First of all, if at a chosen point x the charge density is zero, you could use the Laplace eq. (plus some boundary conditions), if not you could use the poisson eq. (plus some possible boundary conditions). Probably the question also aims at computing the electrical field inside the cylinder where indeed poisson eq. would be needed. Second: you could also use the method of mirror charges, no need to solve a PDG. – Frederic Thomas Jan 18 '17 at 14:16
• how am I supposed to use the method of image charges for a cylindrical conductor? – Chandrahas Jan 19 '17 at 13:11

## 1 Answer

Always use Poisson's equation. That is the general formula that will hold in E&M (in the classical Maxwell formalism). However, it will simplify to Laplace's equation if you are trying to solve the Poisson equation in a region of space where there is no net charge density at any point.

• When we are talking about the charge density, does it mean the charge density as a function of a position vector (x,y,z) or the total charge density. (Total charge enclosed divided by the total volume of the region we are calculating it in.)??? – Chandrahas Jan 19 '17 at 13:06
• In Poisson's equation for the $\mathit electric$ $\mathit potential$, the charge density that you use there is the $\mathit total$ charge density as a function of (x,y,z) (assuming you are working in real space, and not the fourier transform.) – Ben S Jan 19 '17 at 16:47
• In order to make it clear: The "total charge enclosed divided by the total volume of the region we're calculating" is the average charge density of this region. Total density only makes sense if you consider that the given charge distribution consists of free electrons having a density $\rho$ and bound electrons due to el.polarization having a density $\rho_P$. Then $\rho_{total}(x,y,z)= \rho(x,y,z) +\rho_P(x,y,z)$. Poisson eq. says: $\Delta\phi(x,y,z) =4\pi \rho_{total}(x,y,z)$ (in cgs-units). – Frederic Thomas Jan 24 '17 at 14:27