How does a current cause a voltage?

Question

In most textbooks I've read, the explanation of Ohm's Law begins by assuming that there is an external voltage applied to a resistive (Ohmic) device. From there, it is explained that this voltage creates an electric field that causes electrons to flow. As the voltage is increased, the electrons will move faster, creating a larger current in a linear manner,...and you end up with $V=IR$.

This is fine for the situation given -- if there is an applied voltage, the current can be computed with the above. From there, though, I find that most textbooks make the jump to a similar, yet slightly different, situation without much explanation. That is, textbooks will have problems where, say, $1A$ is being supplied to a resistor via a current source and you have to find the voltage across the resistor. This is where I am getting a little confused -- I understand how a voltage can cause a current, but why does a current necessarily cause a voltage (that still satisfies $V=IR$)?

I am asking this because I have been reading about semiconductor physics. To my understanding, semiconductor currents can result from drift and/or diffusion. In the case of diffusion, the source of the current is not a voltage, but rather a concentration gradient.

So, returning to the previous example, if there is $1A$ of current flowing through a resistor, how do we know that the "source" of this current is an electric field (i.e. a voltage) and not just a concentration gradient? If it was just a concentration gradient, wouldn't the voltage across the resistor be $0$ ("violating" Ohm's Law)?

I've been trying to justify this to myself. One thing that comes to mind is perhaps it has to do with unique solutions to Maxwell's Equations. That is just my guess, but I was wondering if there is a better explanation.

EDIT: I think the beginning of Chapter 7 in Griffith's Electrodynamics has a pretty good explanation of what I am getting at. Griffith's explains that, for most substances, $\vec{J}=\sigma \vec{f}$, where $\vec{f}$ is any force per unit charge (e.g. electrical, chemical, gravitational). In other words, just because a certain current is flowing through a resistor does not necessarily imply that there is a voltage across it (i.e. there could be a gravitational potential difference instead, theoretically). I think this pretty much answers my question, so I will close it.

Answer 1

The key point here is that a resistor can handle both causal forms of ohm's law: If we assert a fixed voltage as an input the resistor returns a current as an output, or if we assert a fixed current as an input, the resistor returns a voltage drop as an output.

For example, think of an ideal current source as a positive displacement water pump that runs at whatever flow rate we desire. We connect its inlet to a bucket of water and its outlet to a garden hose. We turn the pump on and measure the pressure drop between the pump outlet and the end of the hose, and get the equivalent of ohm's law: the pressure drop is equal to the flow rate (which is asserted by the pump) times the flow resistance of the hose. In this case we get a pressure drop in response to a flow rate i.e., we assert a current and obtain a voltage as an output.

Now note that in our garden hose example, we can pinch the hose to increase its resistance to flow, but since the pump asserts a fixed flow rate into the hose no matter what, the pressure drop across the resistance increases. Similarly, for an unpinched hose if we turn up the flow rate knob on our positive-displacement pump, the pressure drop across the hose increases.

Answer 2

Another tack.

Why is Ohm's law called a "law"? Because the data show that it is always obeyed. When some function or attribute is called a law in physics, it means that within that mathematical framework it is the axiom needed to be able to model the data mathematically. It means that it has been observed experimentally that the data always obey this law, period. This is more evident in the laws that Maxwell's equations assume as axioms in order to extract from general differential equations , those solutions that obey the laws that were found to be always "obeyed" by the data.

This is where I am getting a little confused -- I understand how a voltage can cause a current, but why does a current necessarily cause a voltage (that still satisfies V=IR)?

In this simple classical case of having to solve algebraic equations, it means that Ohm's law can be used safely when two of the variables are given to find the third. It is a law, which means that there has been no measurement or observation that falsifies it, within the classical framewor.

When one goes to the underlying framework of atoms and molecules and quantum mechanics, one can see the derivations of ohm's law within the more rigorous framework:

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A microscopic view suggests that this proportionality comes from the fact that an applied electric field superimposes a small drift velocity on the free electrons in a metal. For ordinary currents, this drift velocity is on the order of millimeters per second in contrast to the speeds of the electrons themselves which are on the order of a million meters per second. Even the electron speeds are themselves small compared to the speed of transmission of an electrical signal down a wire, which is on the order of the speed of light, 300 million meters per second.

there is further analysis in the link