Biot-Savart law as a tool to calculate the matrix of inductances and the coupling coefficient of two coaxial solenoids with arbitrary cross sections

© The Author(s) 2015. The Biot-Savart law is a basic topic in teaching of electromagnetism in undergraduate courses. However, its applicability in textbooks is reduced, focused on a very specific set of problems. Generally, the student considers that this law does not completely solve some major problems in electromagnetism. One of them is the determination of the magnetic field inside and outside of an ideal solenoid of arbitrary cross section. In this article, we carry out the calculation of this field in a didactic and simple way by employing the Biot-Savart law. We obtain that the magnetic field inside of the solenoid is uniform and independent of its cross section. Besides, we directly determine that the outer field is equal to zero. These results allow us to introduce the expression of the matrix of inductances and the coupling coefficient of two coaxial solenoids with arbitrary cross sections.