Why are potential drops concentrated at resistive elements?
Solution 1
That the potential drop across an ideal (resistivity $\rho = 0$) wire is zero is a consequence of Ohm's Law: $$\mathbf{E} = \rho\,\mathbf{J} = \mathbf{0} \Rightarrow \Delta V = \int{\mathbf{E}\cdot\,d\mathbf{l}} = 0.$$ ($\mathbf{J}$ is current density, which, if you aren't familiar with it, is essentially a current per unit area.)
In a nonideal circuit element, on the other hand, $\mathbf{E} \neq \mathbf{0}$ (for a nonzero current), making the potential drop nonzero. The most important case is that of a homogeneous cylindrical chunk of material (length $L$, crosssectional area $A$). If we set up a uniform current density along the cylinder, the potential difference across it is $$\Delta V = \int{\rho\,\mathbf{J}\cdot{\,d\mathbf{l}}} = \rho JL = \frac{\rho L}{A} I.$$ That is, the voltage is proportional to the current. The constant of proportionality, as you probably know, is called the resistance.
It might also be helpful to you to think about what happens at the interface between two conductors of different resistivity. Consider two connected wires, one of resistivity $\rho_1$ and the other $\rho_2$, with $\rho_1 \neq \rho_2$. When the system is in steady state, the current density just to the left of the interface must match that just to the right otherwise the surface charge at the interface would change with time, as either more or less charge would be flowing in than flowing out, in a unit time. That is, $\mathbf{J}_1 = \mathbf{J}_2 \equiv \mathbf{J}$. Hence $$\mathbf{E_2}  \mathbf{E_1} = (\rho_2  \rho_1)\mathbf{J}.$$ Gauss's Law with a pillbox at the interface as the surface then yields $$\sigma = \epsilon_0(E_2  E_1) = \epsilon_0 J(\rho_2  \rho_1),$$ where $\sigma$ is the surface charge density at the interface. So a surface charge accumulates at the boundary between materials of different resistivity in a circuit.
Solution 2
the potential, to my knowledge, is purely dependent upon location in the field and the magnitude of the charges
That is correct.
What happens is that when current is passed from a highly conductive material to a resistive material there are “surface” charges formed at the boundary between the two different materials. This charge distribution produces a greater E field in the resistive material, where the potential drop is sharper. The relationship between the resulting E fields and the charge distribution is as expected from Gauss’ law.
Solution 3
Lumped elements (including resistors) are idealized models which allow you to deal with electronics without solving Maxwell's equations. Of course, they don't exist in reality: the E field doesn't change abruptly next to a resistor, and there's no such thing as a pure resistor to begin with: it will have some inductance and stray capacitance as well.
Think about rigid bodies in mechanics for comparison: how could they generate any forces on collision if they don't deform? Well, we just assume they do, and it turns out that such simplified models are still useful in some cases.
Resistors work in the same way: if you can neglect all the "parasite" parameters in the electric circuit, you can pretend the field is really concentrated inside, and your model will still predict reality quite well.
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Updated on October 26, 2021Comments

P0W8J6 about 2 years
I do not understand why potential differences in the electric field in a circuit are concentrated at resistors when the potential, to my knowledge, is purely dependent upon location in the field and the magnitude of the charges. Why is the decrease in potential concentrated at resistive elements, rather than decreasing linearly with distance travelled through the circuit?
Similar questions have been asked already on this platform but none have alleviated my confusion.

Peter Cordes about 2 yearsFor similar reasons why gravitational field strength would jump outside vs. inside a hollow shell of lead, but be pretty constant all through an air gap, for a spherical (planetsized?) object made up of different layers. Gravity strength only varies in a simple way if your planet is uniform density.

OverLordGoldDragon about 2 yearsUnfortunately none of the responses are legitimate, nor are you likely to find any on the web. I suggest your own investigation, especially into batteries.


P0W8J6 about 2 yearsSo, if I understand correctly, potential does not decrease linearly with distance through the circuit because of the specific charge distribution around resistive elements?

Dale about 2 yearsYes, the surface charges both on the outside of the conductors and between different conductor materials forms and directs the potential distribution

P0W8J6 about 2 yearsThanks for the response. With regards to your answer, I still don’t understand why the potential of the field shouldn’t be independent of the resistance of the circuit elements. Sure, I get that charges will not lose potential energy in an ideal wire but surely the potential of the field must decrease since you are moving through the field, just like gravitational potential energy decreases linearly as you fall in the field.

Andrew about 2 yearsIt might be worth also including in your answer the case of a constant but nonzero resistivity, to contrast a resistor with an ideal wire.

P0W8J6 about 2 yearsThanks for your responses. Could I just ask why potential decreases linearly as you descend in a gravitational field, rather than having sharper potential drops at certain points as in the case that you describe in you answer?

Dale about 2 yearsNot in comments. You would need to open a new question for that

DGAPhysics1 about 2 years@P0W8J6 The point is that the electric field is zero in the parts of the circuit that are ideal, so the potential difference across themwhich is, after all, the line integral of the electric fieldis zero. Your analogy to gravitation is incorrect because, in that case, the gravitational field is nonzero in the region of interest.

DGAPhysics1 about 2 years@Andrew Good idea. I edited my post.

Dale about 2 years@DGAPhysics1 nice answer +1. I actually like it better than my own answer!