The behavior of elongation flows for a liquid crystal polymer (LCP) solution under art electric field was theoretically studied using the Doi model. The behavior and the alignment of the nematic phase were found to be strongly dependent upon the mutual interactions between external fields such as the electric and flow fields. The effects of these interactions on the alignment were thoroughly examined in this study. The orientation of the nematic phase exhibits the typical behavior of the Freedericksz transition where an abrupt change in the orientation occurs at the applied electric field above a threshold value. The results from the normal stress difference on the LCP solution indicate that this difference is attributed to the Maxwell stress effect and varies with respect to the flow direction. A dramatic reduction in this difference was found when the director of the nematic phase is switched from a direction parallel to the flow to a direction perpendicular to the flow.