Abstract: In this contribution we analyze a typical 3-D optimal design problem in conductivity which consists in seeking the optimal layout of two materials in a given design domain by minimizing the L2-norm of the electric field under a constraint on the amount on each material that we can use. We propose a variational reformulation using a characterization of the three-dimensional divergence-free vector fields and using gradient Young measures as a main tool, we can compute an explicit form of the constrained quasivonexification of the cost density. This result is a natural extension to the 2-D case, but however, the characterization of the divergence-free vector fields introduces a certain nonlinearity in the problem that needs to be addressed properly.