Understanding voltage as a relative measure between two bodies

Question

Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points, which (in a static electric field) is defined as the work needed per unit of charge to move a test charge between the two points. (from Wikipedia)

What exactly is a point in this definition? The voltage of a cell can be measured for example and a cell is not exactly what I intuitively understand to be a point. Let's say I have such a cell with an electrical potential $V_m$ of $-80mV$ (is this the same as saying the cell has a voltage of $-80mV$?).

Am I right then that the voltage is a relative measure between the cell and the extracellular fluid? And is the voltage of this fluid then just the symmetric inverse, i.e. $V_{extra}=80mV$? How then can the voltage be measured between two bodies, because the concentration of charged ions might not be uniform across the fluid (or across the cell)? Could the voltage also be measured between two organelles within the cell or between the entire human body and the air surrounding it?

Answer 1

When we say that "point xxx is at +30V", it means "voltage at point xxx is 30V greater then the one at point yyy", where yyy is a second point implicit or explicit in the phrase.

When it is said that inner parts of a cell are at -80mV, we say "any point (of interest) inside the cell is at a voltage 80mV lower that any point in the fluid surrounding the cell". Not all point at the cell, nor point surrounding it will have exactly the same voltage difference, however, we assume that, for the volumes of interest, the divergence can be ignored.

Recall cell voltages are usually due to electrochemical gradients, differences in the concentrations of some ions at the two size of a membrane creates a difference in voltage. Equations as Nernst or Goldman equations allow calculate the voltage given the concentrations of ions at both side of the membrane.

Ion pumps, ... changes the ion concentrations, producing a change in the voltage of the cell (in reference to volume external to it).

Answer 2

What exactly is a point in this definition?

A point is simply some location in space.

Let's say I have such a cell with an electrical potential $V_m$ of $-80mV$ (is this the same as saying the cell has a voltage of $-80mV$?).

In the case of a voltaic cell (battery), the voltage of a cell is the potential difference between the two terminals of the cell. Therefore, only if the positive terminal of the cell has been assigned an electrical potential of zero volts will the negative terminal electrical potential of -80$mV$ be the same as the voltage of the cell. Keep in mind that the electrical potential at a given point is only the same as the potential difference (voltage) between two points if one of the points is assigned an electrical potential of zero volts, a completely arbitrary decision.

With regard to the living cell in a human body, which your second paragraph appears to be discussing, I can't answer as I know nothing about neuroinformatics that you mentioned in your comments.

But the configuration of the charge between two points does necessarily prevent one from measuring or calculating the voltage between the two points. The calculation of the voltage $V$ would becomes more complicated due to the complex nature of the electric field $\vec E(r)$ in the space between the two points. In general, since the electric field between two points is the gradient of the potential between the points, the calculation involves solving the following integral between the two points

$$V=\int_1^2\vec E(r) \cdot d\vec r$$

Where the vector $\vec r$ points from point 1 to point 2, and $\vec E(r)$ is the electric field as a function of $r$ between the two points. I have no idea what that is in a living cell.

In any case, I hope this helps at least in connection with your first paragraph in understanding the difference between voltage and electrical potential.