Circuit theory is an extremely important basic theory in electrical and electronic sciences and technologies. Over more than a century, researchers have come to the conclusion that a fundamental law of circuits needs to satisfy the following three conditions: (1) Independency. It must be able to solve independently the basic problems of general solutions to the distribution of current and voltage in circuits. (2)Fundamentality. It cannot be derived from circuit theory and it must be the starting point for the establishment of circuit theory; it deduces the problem relevant to circuit theory by using purely logical inference, and establishes circuit theory into an independent deductive system. (3) Applicability. It must be widely applicable to all spheres of circuits,which includes sinusoidal steady-state linear and nonlinear networks, non-sinusoidal steady-state linear and nonlinear networks, transient-state processes, etc. From all networks to which the fundamental law of circuits applies, sinusoidal steady-state linear network is chosen as the most basic one to demonstrate that the two independent equations of circuits in integral form derived from Maxwell equations are able to meet these three conditions. Consequently, it is believed to be the fundamental law of circuits newly recognized today. This paper also makes the initiative to establish a circuit theory by which the basic rules of electromagnetic field govern the circuits, and the unity of electromagnetic fields and circuits is achieved.