# Voltage and Resistance definition

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.

• – MarcoCiafa Jun 4 at 1:11

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 $$I$$
• 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 $I = \int_{S} \,\mathbf{J}\cdot\hat{n}\,dS > 0$), $V = -\Delta\phi$? – MarcoCiafa Jun 4 at 1:07
• Yes, because current goes in the direction of decreasing $\phi$ – Dale Jun 4 at 1:11