The formula to calculate a variation of optical path length of single crystal by the electric field was derived by this study. The formula was applied to $Bi_{12}SiO_{20}$ single crystal. The results are as follows. In case of the applied electric field in the body diagonal direction and the passing light along the same direction, the variation of optical path length had the largest value. The symmetry of the space distribution of optical path length satisfied $E3C_2;8C_3$, the set of elements of the symmetry of $Bi_{12}SiO_{20}$ single crystal. The property which gave the largest influence to the variation of optical path length is the strain of length by the Inverse piezoelectric effect. The second influence, is the variation of the refractive index by the electro-optic effect. The variation of optical path length by the inverse piezoelectric effect and by the electro-optic effect have a reverse sign each other.??????????