of . Mathematically, since is perpendicular to as and are along the same direction. Now we can see that, for the conductive loop, emf is same to the time-derivative of the magnetic flux through the loop except for the sign on it. Therefore, we now reach the equation of Faraday's law (for the conductive loop) as where . With breaking this integral, is for the transformer emf (due to a time-varying magnetic field) and is for the motional emf (due to the magnetic Lorentz force on charges by the motion or deformation of the loop in the magnetic field). Exceptions It is tempting to generalize Faraday's law to state: If is any arbitrary closed loop in space whatsoever, then the total time derivative of magnetic flux through equals the emf around . This statement, however, is not always true and the reason is not just from the obvious reason that emf is undefined in empty space when no conductor is present. As noted in the previous section, Faraday's law is not guaranteed to work unless the velocity of the abstract curve matches the actual velocity of the material conducting the electricity. The two examples illustrated below show that one often obtains incorrect results when the motion of is divorced from the motion of the material. One can analyze examples like these by taking care that the path moves with the same velocity as the material. Alternatively, one can always correctly calculate the emf by combining Lorentz force law with the Maxwell–Faraday equation: where "it is very important to notice that (1) is the velocity of the conductor ... not the velocity of the path element and (2) in general, the partial derivative with respect to time cannot be moved outside the integral since the area is a function of time." Faraday's law and relativity Two phenomena Faraday's law is a single equation describing two different phenomena: the motional emf generated by a magnetic force on a moving wire (see the Lorentz force), and the transformer emf generated by an electric force due to a changing magnetic field (described by the Maxwell–Faraday equation). James Clerk Maxwell drew attention to this fact in his 1861 paper On Physical Lines of Force. In the latter half of Part II of that paper, Maxwell gives a separate physical explanation for each of the two phenomena. A reference to these two aspects of electromagnetic induction