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A solution of Maxwell's equations is obtained in a resonant cavity with a center post of arbitrary electrical properties. The solution gives the dielectric coefficient and the conductivity of the center post in terms of the natural frequency and Q of the cavity. The theory is of particular use in the study of semiconductors where perturbation theories are of little value. It is shown that a transition from a cylindrical mode to a coaxial mode occurs as the conductivity of the center post is varied. This transition occurs for a relatively small change in conductivity. The present results are compared with those of perturbation theory, and it is shown that the latter are valid over a greater range than the conditions imposed in their derivation indicate.