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Consider a wire of lenght $L$ and transversal area $A$ that it isn't an ideal conductor, but follows Ohm's Law. After a few computations we have

$$-\Delta\phi = \rho\frac{L}{A} I $$

where $\rho$ is electrical resistivity. Note that $\Delta\phi < 0$.

My question is if voltage and resistance are defined as:

1) $V =-\Delta\phi$ (positive) and $ R = \rho\frac{L}{A}$ (positive)

or

2) $V = \Delta\phi$ (negative) and $ R =-\rho\frac{L}{A}$ (negative).

Supporting 1): Voltage is the work done by the Electric Field to move a charge from $\mathbf{A}$ to $\mathbf{B}$. And since the difference of potential is the work done by an external force against the Eletric Field from $\mathbf{A}$ to $\mathbf{B}$, I think this definition makes sence.

Supporting 2): I've read in some places (for example, this answer) that voltage is the difference of potential. I could have misunderstood what they meant by that. Also, when I was thought this, I've never been explicitly told what was voltage.

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Resistance $R$ is positive. So (2) is incorrect. The sign on $V$ is a matter of convention and depends on which direction you call positive for $I$

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  • $\begingroup$ Thank you for answering! In this example, to keep it simple, if I call I positive in the direction of $\mathbf{J}$ (i.e $\hat{n}$ is in the direction of $\mathbf{J}$ so that $I = \int_{S} \,\mathbf{J}\cdot\hat{n}\,dS > 0$), $ V = -\Delta\phi$? $\endgroup$
    – Gilgamesh
    Commented Jun 4, 2020 at 1:07
  • $\begingroup$ Yes, because current goes in the direction of decreasing $\phi$ $\endgroup$
    – Dale
    Commented Jun 4, 2020 at 1:11
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The word voltage is used because electrical potential is measured in volts. Similarly mileage for distance travelled by a car. Potential difference is a technically more correct term than voltage.

Your definition 1 is correct, though it is sometimes called emf (curly E - I don't know the MathJax) to distinguish it from a measured voltage.

Unless you are careful the + and - signs can cause problems. The difference between two numbers is the larger minus the smaller, so it is positive, and in the same way the potential difference across a battery or a resistor is normally stated as a positive number. Usually if you follow the direction of the current an emf is a rise in potential. The exception might be if there is another potential source in the circuit. There is a drop in potential in the direction of the current through a resistance.

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