I've done my B.S. in Electrical Engineering as well as mathematics but I'd like to get a proper, or complete history of Maxwell and the history of his derivation of the equations and the newness of the methods he employed. Freeman Dyson's description of how mathematics parted from physics in the 19th century has me looking into what details he was describing. As always your input is greatly valued.
Just quickly adding some more references for you. The analytical theory of EM starts with Biot and Poisson at the turn of the 19th century. The works by Poisson were particularly influential to George Green, who in turn was influential to Maxwell. Faraday was also critical to Maxwell's conceptions of the electromagnetic field, particularly his first few researches on electricity and magnetism, and the last few (if I recall correctly). They're good reads, and readily available.
I wrote a translation of Poisson's first memoir on electricity, posted on my website here, and Biot's paper is there too. Green's works are available in English online. You could also look into Ampere's writing and contemporary reviews of it. That'll set you up for reading Maxwell, and understanding the views he expresses.
The most direct resource that I personally have experience with would be Simpson's Maxwell on the Electromagnetic Field, which is an annotated walk-through of three of Maxwell's most important papers. It's written for a non-technical audience, but it doesn't shy away from details; it's filled with notes and context and I like it quite a bit.
Then after reading Maxwell's first few papers, you can read his Treatise, which is the master work. You'll learn things about electromagnetism there that they stopped teaching a long time ago, but it's still very valuable. Good luck!
Biot, J-B. Sur un problem physique, relatif a l'electricite (1801). Link.
Poisson, S-D. Memoire sur la distribution de l'electricite a la surfaces des corps conducteurs (1812-13). Link.
Green, George. An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828). Link.
Faraday, Michael. Experimental Researches in Electricity (1831-54). Series I, II, III, IX, XI, XIV, XIX, XXVIII, XXIX.
Assis & Chaib. Ampere's Electrodynamics: Analysis of the Meaning and Evolution of Ampere’s Force between Current Elements (2015).
Simpson, Thomas K. Maxwell on the Electromagnetic Field (1997).
Maxwell, J. C. On Faraday's Lines of Force (1855). Transactions of the Cambridge Philosophical Society, Vol X. part I.
Maxwell, J. C. On Physical Lines of Force (1861). Philosophical Magazine, Vol XXI.
Maxwell, J. C. A Dynamical Theory of the Electromagnetic Field (1864). Royal Society Transactions, Vol. CLV.
Maxwell, J. C. A Treatise on Electricity and Magnetism (1873).
- Pierre Duhem, The Electric Theories of J. Clerk Maxwell, trans. of 1902 Les théories électriques de J. Clerk Maxwell (2015 re-typset ed.), vol. 314, Boston Studies in the Philosophy and History of Science (Cham: Springer International Publishing, 2015).
- Sarkar, Tapan. 2006. History of Wireless. Hoboken, N.J.: Wiley-Interscience.