It has been observed that the exposure of dielectrics to electron beams can produce electric fields of sufficient magnitude to cause dielectric break-down (see e.g. Refs. 1 and 2). It is the purpose of this investigation to calculuate the electric fields for two of the geometric shapes of common interest: spherical shells and cylinders. This report will summarize the work done to date on spherical dielectric shells with an inner and outer conducting surface which are traversed by an electron beam through the sphere's center. Future work will deal with a spherical shell when the beam is displaced from the sphere's center and with cylinders.
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