Given total charge, how to calculate the surface-charge distribution Suppose you given conductors $L_i$ with given geometry in space and the information that the conductor $L_i$ has the total charge $Q_i$ ($i = 1,\dots,n$). Suppose further that there are no additional charge density outside of the $L_i$.
I want to calculate (numerically) the resulting electrostatic surface charge distribution and the electric field. 
What is the differential equation I have to solve? Is the solution unique with the information above?
 A: The differential equation is Laplace's equation:
$$
\nabla^2 V=0
$$
with the boundary condition: surfaces of conductors are at a uniform potential.  this gets you $V$, from this it is easy to get $E$ and from $E$ use Gauss' law to get the charge distribution:
$$
\sigma=E\epsilon_0
$$
In general, however, this is a very computationally heavy problem.
A: We can use method of moments here. First, we divide up the conductor surfaces into $m$ small patches where the $i$th patch has $q_i$ charge. Next, we can compute the electric potential at each patch as a function of the other patches. That is, we can write in matrix form $\vec{v} = A\vec{q}$ where $\vec{v}$ is vector with length $m$ where $v_i$ represents the potential at the $i$th patch, $\vec{q}$ is the list of charges, and $A_{ij}$ is the contribution of patch $j$ to the the potential of patch $i$, which can be computed directly from the choice of patches. This is a linear system with $m$ equations and $m + n$ unknowns ($m$ unknown charges and $n$ different potentials for each conductor). However, we also have $n$ constraints from the total charge constraints on each conductor. We can therefore solve for this linear system to get the charge distribution of each conductor.
