Faradays law and the first Maxwell equation

Early investigators of electric and magnetic fields observed that when the magnetic induction vector B changes with time throughout a surface S bounded by a contour L, an electromotive force S exists along that contour with an intensity  

As is well known, the electromotive force can also be presented as  

This expression can be interpreted in a natural sense as follows a change with time of the magnetic flux, ip, gives rise to an electric field. This phenomenon was first observed and reported by Faraday and has been called electromagnetic induction. The relationship between the electric field and the rate of change of the magnetic flux, as described by eq. 1.180, is one of the most fundamental relationships in physics. 

By convention, the electric field due to electromagnetic induction is called inductive electric field and noted emphasizing the origin of this particular component. One can rewrite eq. 1.180 in the following form  

It is a basic fact that a change in magnetic flux with time gives rise to a specific electromotive force. To determine the field however, additional information must be provided. 

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