Conservation theorems are central to many aspects of physics: they often form the central reasoning principles for new observations, they provide checks on new predictions, and they appear to be obeyed regardless of system and scale. You might not have heard them called conservation theorems before, but you have used them. In mechanics, these theorems manifest themselves as the three fundamental principles (for momentum, energy, and angular momentum):

$$\Delta \vec{p}_{sys} = \vec{F}_{ext} \Delta t$$ $$\Delta {E}_{sys} = W_{ext} + Q$$ $$\Delta \vec{L}_{sys} = \vec{\tau}_{ext} \Delta t.$$

Electromagnetism is consistent with these fundamental principles (as you will see), but now that matter has charge, we bring a fourth fundamental principle to the party,

$$\Delta Q_{sys} = I_{ext} \Delta t.$$

These principles are referred to as “conservation theorems” because the describe how properties of a system will change and, in principles, under what conditions those properties will not change (i.e., how they are conserved),

$$\Delta \vec{p}_{sys} = 0\,\mathrm{when}\, \vec{F}_{ext} = 0 $$ $$\Delta {E}_{sys} = 0\,\mathrm{when}\, W_{ext} + Q = 0$$ $$\Delta \vec{L}_{sys} = 0\,\mathrm{when}\, \vec{\tau}_{ext} = 0$$ $$\Delta Q_{sys} = 0\,\mathrm{when}\, I_{ext} = 0 $$

Linear and angular momentum conservation in an electromagnetic system are outside the scope of this course. This is because to truly understand the relationship between these and the electromagnetic field, we must develop an understanding that the electromagnetic field can have linear and angular momentum. That's right, the field itself has momentum that can push physical objects or twist them. This might seem very strange, but it is definitely the case that the electromagnetic field itself can have both.

Consider an example that is common in astrophysics. A star is going through fusion pushing gas outward from the core. In addition, light is carried outward. This is complicated process, but the gas and light run into material in front of them as they move towards the stellar surface. These pushes by the gas and light cause a pressure on the material in front of them; pushing them outward. However, the gas in front of the outward moving gas and light is gravitationally attracted to any matter behind it. This careful balance of the gravitational pressure, gas pressure, and radiation pressure (the momentum imparted by collisions of electromagnetic radiation with material) determines the size, temperature, and brightness of the star. Stellar formation and evolution is vast research topic, but the point is that without that radiation pressure from the momentum carried by the electromagnetic radiation, the star could collapse under it's own gravity – in fact, this is what happens in core collapse supernovae!