  We introduce the concept of emergent electric field. This is distinguished
from the fundamental one in that the emergent electric field directly appears
in observations through the Lorentz force, while the latter enters the phase
space as the canonical momentum of the electromagnetic field. In Hamiltonian
classical electromagnetism this concept naturally appears after introducing the
topological $\theta$ term. Furthermore, we show that in the spherically
symmetric model the concept of emergent electric field allows us to formulate a
modified theory of electromagnetism that is otherwise impossible. The relation
between the fundamental and the emergent electric fields is derived from the
imposition of general covariance of the electromagnetic strength tensor, which
is a nontrivial task in the canonical formulation the modified theory is based
on. We couple this theory to emergent modified gravity, where a similar
distinction between spacetime and gravity is made such that the spacetime,
which defines the observable geometry, is an emergent field composed of the
fundamental gravitational field. In this more encompassing emergent field
theory coupling gravity and electromagnetism, we show that the spherically
symmetric model contains a nonsingular black hole solution where not only
modified gravity but also modified electromagnetism is crucial for a robust
singularity resolution and to avoid the existence of (super)extermal black
holes.
