The work of Oliver Heaviside and Laplace put the electrical theories in a firm footing.
Heaviside invented an operational calculus for solving differential equations arising out of electrical network analysis, which was justified rigorously later by Laplace Transforms(but which makes full sense only incorporating the theory of distributions).
This might not seem important enough historically. But, all power generation, motors, the light you have in your room, and indeed all uses of electricity were able to be set up properly thanks to the work of these people, and the midnight oil they burned. We wouldn't have computers or MO without electricity distribution everywhere, for instance.