An Electrooptic Analog for the Solution of the Laplace Equation

Laplace’s equation occurs frequently in mathematical physics for problems relating to fluid flow, heat transfer, and so on. For some simple cases, the boundary-value problem can be solved; but more often, the differential equation proves intractable, and numerical analysis or experimental methods are used. The electrooptic analog is an experimental method based upon the fact that an organic dye solution becomes birefringent in an electric field. This effect enables one to determine voltage gradient throughout a two-dimensional field. The boundary conditions most readily applied are prescribed constant values of the electric potential φ on conducting segments of a boundary and ∂φ/∂n = 0 on insulated segments of a boundary. With the known conditions on the boundary and the potential gradient found from experiment, the problem is solved. This analog can be used for all physical problems for which the boundary conditions are applicable, and which satisfy Laplace’s equation.