Choose electrostatic boundary data consistently

Dirichlet data assign a potential value at each boundary point relative to a chosen reference. Neumann data assign the outward normal derivative there. Since electric field is minus the gradient, positive outward electric field corresponds to a negative outward potential derivative. On the left and right ends of an interval, the outward normals point in opposite directions. The same Cartesian field therefore gives normal components of opposite signs at those ends.

A conductor held at a prescribed voltage is an equipotential boundary whose surface charge is determined by the solution. An isolated conductor with known total charge is also equipotential, but its potential is initially unknown. Its total charge provides an integral constraint, not the pointwise Neumann data. Substituting a uniform charge density merely because the total charge is known changes the problem unless symmetry independently warrants uniformity.

Integrate Poisson’s equation over the region and use the divergence theorem. The integral of the prescribed normal derivative must equal minus the enclosed charge divided by permittivity. This compatibility condition is necessary for a pure Neumann solution. For example, zero derivative on a closed boundary is incompatible with a nonzero net interior charge. No choice of an additive constant can repair that mismatch because constants affect neither derivatives nor flux.

For compatible data in a connected bounded domain, the same squared-gradient argument used for Dirichlet uniqueness shows that two solutions differ by a constant. Now the boundary term vanishes because the difference derivative is zero, rather than because its value is zero. Specify a mean potential or one reference value to choose that constant. The electric field is already unique before this gauge choice, so the remaining ambiguity is not a physical uncertainty in force.

One can prescribe value data on one portion of a boundary and derivative data on the remainder. With a Dirichlet portion of nonzero boundary area, the additive constant is fixed and the usual homogeneous difference argument gives uniqueness. Conversely, prescribing both value and derivative independently on every boundary point generally overdetermines an elliptic problem. A measured pair may be consistent, but it cannot be chosen arbitrarily.

Use the vector lab to separate a vector from its normal projection. Reset and set angle to 0° and magnitude to 2; predict an x component of 2. Reset and set angle to 60° and magnitude to 2; predict an x component of 1. If the normal is the positive x direction these are outward field components; reversing the normal reverses their signs. This projection instrument has no closed surface, charge integration, or boundary-value solver.