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I'm wondering if there is a way to calculate voltage drop across part of a system if you know resistance and the potential difference of the entire system as well as its subparts but don't know the current. It seems like there should be a way considering that voltage is joules per coulomb so voltage drop shouldn't depend on how many coulombs are flowing through the system.

Let's say you have an electrical distribution wire with 40,000 volts relative to ground, but an unknown current. Let's say that the insulated wire gets grounded somehow (by a tree falling on it, etc.) You would know that the initial voltage is 40,000V and the final voltage is 0V since the current flows to the ground. Could you figure out how much voltage was dropped by the insulation on the wire and then how much voltage was dropped by the tree, assuming that you know the resistance of both the insulation and the tree, but not the current flowing through the system? If so, how?

Note: You can't figure out the current using ohm's law because the current is limited by transformers.

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You might not be able to use Ohm's Law directly to solve for the current of the system (as some fuse will almost instantaneously trip so that there isn't a short in the line), but you could use it indirectly to analyze the situation.

In this circumstance, it would in fact be true that the instantaneous current in both the tree and insulation of the wire are the same, as this is an example of a single-loop "circuit" (used in a loose sense of the word), and by Kirchoff's Junction Rule, the current everywhere in a single-loop circuit is the same in both the tree and the insulation, though unknown in this problem.

We can use Ohm's Law because the current in the tree, $i_t$ is equal to the current in the insulation, $i_i$. Therefore, since $V=iR$, $$\frac{V_t}{R_t} = \frac{V_i}{R_i}$$ This equation can be solved for the fraction $\frac{V_t}{V_i}$ and set equal to $\frac{R_t}{R_i}$, and in your question, you stated that you knew, or could find out, the resistances of both materials. Now you know the fractional electric potentials for both media.

Thus, if the total electric potential is known, $V_{total}$, you can find the potential drop in each one using the equation $V_{total} = V_t + V_i$, and viola, you now know the potential drop in each insulator!

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