We present an analytic solution for the B1 field produced in a gapped toroidal cavity resonator designed as a probe for high field MRI. This resonator supports standing TEM waves, so its electric and magnetic fields are identical to those produced by a stationary planar current source with the same (constant) cross-section multiplied by a complex exponential propagation factor. An explicit expression for the field may therefore be found by solving Laplace's equation for the static potential, which is accomplished with a two-dimensional logarithmic conformal transformation algorithm. The equipotential curves are also the contours of the field strength B, and the B (vector) field at any point is directed along the contour passing through that point. With this information, we construct the solution by computing the angle made by the equipotential curve with the horizontal axis at each point, using this angle to analyze the B field into its x and y components, and adding the contributions from the current sources to obtain the magnitude and direction of B at each point in the region of interest. Some proposed extensions of this algorithm are also discussed.