Estimating and synthesizing the hand's manipulation of objects is central to understanding human behaviour. To accurately model the interaction between the hand and object (referred to as the "hand-object"), we must not only focus on the pose of the hand and object, but also consider the contact between them. This contact provides valuable information for generating semantically and physically plausible grasps. In this paper, we propose an explicit contact representation called Contact Potential Field (CPF). In CPF, we model the contact between a pair of hand-object vertices as a spring-mass system. This system encodes the distance of the pair, as well as a likelihood of that contact being stable. Therefore, the system of multiple extended and compressed springs forms an elastic potential field with minimal energy at the optimal grasp position. We apply CPF to two relevant tasks, namely, hand-object pose estimation and grasping pose generation. Extensive experiments on the two challenging tasks and three commonly used datasets have demonstrated that our method can achieve state-of-the-art in several reconstruction metrics, allowing us to produce more physically plausible hand-object poses even when the ground-truth exhibits severe interpenetration or disjointedness. Our model and source codes are made publicly available at https://github.com/lixiny/CPF.