In 1968, F.L. Shapiro published the suggestion that one could search for an electron EDM by applying a strong electric field to a substance that has an unpaired electron spin; at low temperature, the EDM interaction would lead to a net sample magnetization that can be detected with a SQUID magnetometer. One experimental EDM search based on this technique was published, and for a number of reasons including high sample conductivity, high operating temperature, and limited SQUID technology, the result was not particularly sensitive compared to other experiments in the late 1970's.Advances in SQUID and conventional magnetometery had led us to reconsider this type of experiment, which can be extended to searches and tests other than EDMs (e.g., test of Lorentz invariance). In addition, the complementary measurement of an EDM-induced sample electric polarization due to application of a magnetic field to a paramagnetic sample might be effective using modern ultrasensitive charge measurement techniques. A possible paramagnetic material is Gd-substituted YIG which has very low conductivity and a net enhancement (atomic enhancement times crystal screening) of order unity. Use of a reasonable volume (100's of cc) sample of this material at 50 mK and 10 kV/cm might yield an electron EDM sensitivity of 10 −32 e cm or better, a factor of 10 5 improvement over current experimental limits.