In proper differential geometric notation, the fundamental equations underlying electrodynamics, namely the Maxwell's equations, take the following form:
$$dF=0,$$
$$d*F=J,$$
where $F$ is the electromagnetic field strength, $J$ is the current (for example electrons) and $*$ is the Hodge star operator.
Since the first equation just says that $F\ $ is closed, and since the four dimensional spacetime topology is usually considered to be uncomplicated, by the Poincaré lemma we can introduce a 1-form $$A=\sum_{\mu=0}^3A_{\mu}dx^{\mu},$$
the electromagnetic four-potential, which satisfies
$$F=dA.$$
There are many ways to go from here. I just saw that at the bottom of the Poincaré lemma wikipieda page, they elaborate more on the magnetic aspect of this. As you can see, this is the part which deals with the spatial components of $A$, i.e. the bottom three components. Therefore I think it's worth considering the other extreme here: Simple situations where $A$ has actually only one component, i.e. $A=A_0dx^0\equiv\phi\ dt.$ This is what physicists like to call "the theory of electrostatics". (I'm allowed to talk like this, I'm a physicist myself.) The line that follows
$$F=dA=d(\phi\ dt)=d\phi\wedge dt=-dt \wedge d\phi=dt \wedge \left(-\sum_{\mu=0}^3\frac{\partial\phi}{\partial x^{\mu}}dx^{\mu}\right)\equiv dt\wedge E,$$
might be recognisable. It's the cotangent version of $\vec{E}=-\ grad(\phi).$
How this $\vec{E}$ is related to the exact 1-form $E$, whos components are just the time components of the two form $F$ , is determined by the spacetime metric. The components are identical for in flat spacetime. A remark regarding $E$ or $d(-\phi)$ in this context: You can see that, naturally, here the Poincaré lemma hits again - this time in three-dimensional space, where the Maxwell equations state that $curl(\vec{E})=\vec{0}$. So if you don't consider the full theory and exclude time dependence, then you can start from this relation.
Now define the electrostatic force $\vec{F}:=Q\ \vec{E}$, where $Q$ is an electric charge, together with the usual definition of work $W:=\int \vec{F}\ d\vec{s}$ as well as Stokes theorem (or really just the fundamental theorem of calculus), and your everyday school electrostatics follows at once:
$$W=\int \vec{F}\ d\vec{s}=\int Q\ \vec{E}\ d\vec{s}=-Q\int grad(\phi)\ d\vec{s}=-Q\ (\phi_2-\phi_1)\equiv Q\ U,$$
where the potential difference $U$ is called voltage.
Another interesting point is the introduction of gauges, which reflect that due to $d^2=0$ we find that $F=d(A+d\alpha)$ holds for any $\alpha$. There are several $A$ for only one electromagnetic field $F$. Broad generalizations of this concepts eventually lead to all the other physical theories involving charges, like the theory of quarks. In fact the whole standard model of particle physics, which describes three of the four fundamental forces, takes this route. This then covers basically all of physics, except for gravity. But, oh snap, that's differential geometry as well!
(On that note, all the differential geometric aspects of the beautiful manifolds termed Lie groups would make an equally long post and another good application. Rotation symmetry, forced on us by the spacetime metric model, the related rotation group $SO(3)$, or more specifically it's universal cover $SU(2)$, and its representations ...they introduce the notion of spin, which eventually explains the stability of matter.)
