Re: How do you prove that the magnetizing field H due to a current I is linear?

I could not find the exact equation you cite in your question, although 
there are similar equations for solenoids, for example.  

The magnetic field strength, H, is actually equal to (B - ľo M) / ľo ,
where B is the magnetic field, M is the magnetization (magnetic moment per 
unit volume of a core, like a core of iron inside a solenoid that can get 
magnetized), and ľo is the permeability constant, 1.26E-6 henry/meter.

In any case, these sort of equations regarding magnetic field strength are 
based on the Biot-Savart law:
H=Integral[(Idl X a)/(4*PI*R^2)]. (A/m)


The Biot-Savart Law is like Coulomb's law,  but for magnetostatics. It was 
derived from experimental observations. 

Similarly, the linear relationship between magnetic field strenghth and 
current was also apparently based on experimental observation. With zero 
current, there was no (zero) magnetic field strenghth. As the current was 
increased, the magnetic field strength increased linearly. 

Some useful references that might help to further understand this would be:
1) Halliday and Resnick, Physics (Parts I and II), chapters 34, 35, and 37, 
especially pages 936-940.
2) Skitek and Marshall, Electromagnetic Concepts and Applications, chapter 
8, pages 222-254.