# Meaning of Voltage in Ohm's Law

Ohm's Law states $$V=IR$$

where $$V$$ is te voltage, $$I$$ the current and $$R$$ the resistance.

Since voltage is defined in terms of electric potential, and electric potential defined in terms of electric potential energy, I reason the following:

The electric potential energy of any point $$p$$ must be measured in relation to another point $$q$$, the electric potential of any point $$p$$ should also be measured in relation to aonther point $$q$$. Here the electric potential of $$p$$ is the amount of work necessary to move a unit charge from $$p$$ to $$q$$.

Taking this into account, I wonder if the voltage $$V$$ is the difference in electric potential between both extremes of the circuit, call them $$a$$, $$b$$. The electric potential $$P_a$$ of $$a$$ being the work necessary to move a unit of charge from $$a$$ to $$b$$ following the circuit's path, while the electric potential $$P_b$$ of $$b$$ is the work necessary to move a unit charge from $$b$$ to itself, that is, $$0$$.

Thus, the voltage between $$a$$ and $$b$$ is

$$P_a-P_b=P_a$$

namely, the electric potential of $$a$$. Or put differently, the work necessary to move a unit charge from $$a$$ to $$b$$ along the circuit's path.

Is such an understanding of Voltage correct?

The "zero point" of electric potential can be arbitrary and it does not matter. You could take that point to be at $$b$$, like you say in your post, or you could take it to be at infinity, as it is conventionally done. What does matter is the potential difference between two points, which should be the same no matter what zero point you pick. This is a consequence of the fact that Maxwell's equations are linear, which means you can add and subtract electric potentials.

• If I'm correct in interpreting the distance the work is to be performed to be along the circuit, how could one of the points be taken to be at infinity?
– Sam
Jun 7, 2021 at 8:46

Yes, your understanding is correct but let me clarify a little. The equation you wrote $$P_a - P_b = P_a$$ is valid when you set $$P_b$$ as the reference point to define voltage. If you set a reference point at different points, then (and more generally) $$P_a - P_b = V.$$ Voltage is, as you wrote, the difference in electric potential energy between two points. The electric field is the ratio of the change in the electric potential energy.

• So regardless of which pair of points in the circuit we choose, the voltage between them will be the same?
– Sam
Jun 7, 2021 at 8:31
• This seems confusing. If the voltage is taken between $a$ and point $c$ between $a$ and $b$, then the work necessary to move a unit charge from $a$ to $c$ should differ from the work necessary to move a unit charge from $a$ to $b$.
– Sam
Jun 7, 2021 at 8:38
• The voltage between two points is proportional to the current flowing between those same two points. But the coefficion of proportionality R needs not be the same everywhere in the loop. Jun 7, 2021 at 13:04
• @Leo Sure, because the voltage between $a$ and $b$ need not to be the same as the voltage between $a$ and $c$. Whenever the voltage is different either the current or the voltage can be different. As Marius pointed out, if the current is the same, the resistance can be changed. If the resistance is the same, then the current would be different. Or both can be different. All these are in general. Jun 7, 2021 at 13:39