We present an approach to assess the risk taken by on-road vehicles within the framework of artificial field theory, envisioned for safety analysis and design of driving support/automation applications. Here, any obstacle (neighboring entity on the road) to the subject vehicle is treated as a finite scalar risk field that is formulated in the predicted configuration space of the subject vehicle. The driving risk estimate is the strength of the risk field at the subject vehicle's future location. This risk field is formulated as the product of two factors: collision probability and expected crash energy. The collision probability with neighboring vehicles is estimated based on probabilistic motion predictions. The risk can be assessed for a single time step or over multiple future time steps, depending on the required temporal resolution of the estimates. We verified the single step approach in three near-crash situations from a naturalistic dataset and in cut-in and hard-braking scenarios with simulation and showed the application of the multi-step approach in selecting the safest path in a lane-drop section. The risk descriptions from the proposed approach qualitatively reflect the narration of the situation and are in general consistent with Time To Collision. Compared to current surrogate measures of safety, the proposed risk estimate provides a better basis to assess the driving safety of an individual vehicle by considering the uncertainty over the future ambient traffic state and magnitude of expected crash consequences. The proposed driving risk model can be used as a component of intelligent vehicle safety applications and as a comprehensive surrogate measure for assessing traffic safety.