How can I combine my knowledge of signal processing with electromagnetic physics?
Wow! There's a ton of applications that combine electromagnetics with signal processing, in application areas that are very relevant today. Just think about motors, generators, particle accelerators, Hi-Fi loudspeakers,  rail guns, linear motors, levitating trains, you name it. Energy sector alone is one of the hottest topics in today's world. 

You're very lucky to have found two areas explained to you in so different terms, one with a lot of maths, the other one with less. If they would have been explained more similarly, you wouldn't have so diverse outlook on the field. Combining the two approaches into a practical application is a challenging but joyous experience. 

I would suggest that you take a pragmatic approach to your problem of combining the two: take a look at sensors and actuators that measure and generate electromagnetic fields. They use electricity for the purpose, and electricity has very measurable properties: voltage and current. These can be converted into digital signals by the use of analog-to-digital converters and can be processed. And digital signals can control the power supplies to actuators such as motors using various methods of digital-to-analog conversion. And you often combine both approaches, if you need to monitor the actuator you're controlling. 

As an answer to your final, more precise question, the EM field can be measured with sensors such as Hall sensors in case of DC (non-alternating) field or pick-up coils in case of AC (alternating) field. The signal is converted to numbers using analog-to-digital converters and the digital information is processed using digital circuitry such as microprocessors, custom-made digital logic or other integrated circuits.