Resolving A Paradox in the Teaching of Faraday's Law - [Education Column]

In the classroom development of Faraday's law and its application to transformers, one inevitably obtains a paradox. The magnetic field outside the transformer's core is assumed to be zero in the calculation, while the electric field outside the core must be nonzero if a voltage is to appear in the transformer's secondary. This electromagnetic field external to the core violates Maxwell's equations. We resolve this paradox by finding both the electric and magnetic fields, and using a line integration of the electric field to obtain the voltage in the secondary. We show it to be nearly identical to the result obtained by the traditional application of Faraday's law of induction, based on changing flux in the transformer's core. Justification for the traditional approach is provided. The discussion is of a level typically seen in graduate-level courses, and can provide a good example and challenge for computing the electric fields using vector potentials. With guidance, this paper could also be comprehensible to undergraduates in electromagnetic engineering, particularly those in two-semester EM courses who spend significant time on the elementary integral equations. A discussion of the paradox proposed could be used in most basic EM courses to "wake the students up" and to challenge them to think about Faraday's law, with the paper provided as a proof for the more motivated and interested students in the class. Students may also be asked to verify the results given here through the use of a software package, such as MATLAB