# 2 definitions for voltage, how are they equivalent?

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## Main Question or Discussion Point

So I'm studying electrostatics and I came across to two different definitions of potential difference/voltage (because we're in stationary regimes) and I'm having trouble understanding how the expressions are equivalent.

They are for a voltage between point A and point B

$$U=V_a - V_b =\int_{a}^{b} \textbf{E} \cdot d\textbf{s}$$

and, on the other hand,

$$U= V_b - V_a = - \int_{a}^{b} \textbf{E} \cdot d\textbf{s}$$

How can both of this expressions represent the potential difference between points A and B? Aren't they symmetric?

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Doc Al
Mentor
Not quite sure I understand the question.
How can both of this expressions represent the potential difference between points A and B?
The sign of the potential difference depends on which point you're measuring with respect to:
$V_a - V_b = - (V_b - V_a)$

Not quite sure I understand the question.

The sign of the potential difference depends on which point you're measuring with respect to:
$V_a - V_b = - (V_b - V_a)$
My question is that in both cases the expression is presented as "the potential difference between points a and b". However if I use both definitions will I obtain the same result or will I obtain symmetric results? I'm thinking the later is probably the correct one but now that leaves me with other question: shouldn't the second expression be the potential difference between points b and a?

Mister T
Gold Member
$$U=V_a - V_b =\int_{a}^{b} \textbf{E} \cdot d\textbf{s}$$
What does $U$ stand for?

and, on the other hand,

$$U= V_b - V_a = - \int_{a}^{b} \textbf{E} \cdot d\textbf{s}$$
$V_b-V_a \neq V_a-V_b$ except for the trivial case of $V_a=V_b$.

What does $U$ stand for?

$V_b-V_a \neq V_a-V_b$ except for the trivial case of $V_a=V_b$.
U is the voltage between points a and b.

Exactly that's my doubt, are both expressions correct?

Mister T
Gold Member
$$V_a - V_b =\int_{a}^{b} \textbf{E} \cdot d\textbf{s}$$

and,

$$V_b - V_a = - \int_{a}^{b} \textbf{E} \cdot d\textbf{s}$$

But they cannot both be equal to $U$ at the same time, except when $U=0$.

Tom.G
U=Va−Vb=∫baE⋅ds
U=Vb−Va=−∫baE⋅ds
How can both of this expressions represent the potential difference between points A and B?
U is the voltage between points a and b.
Consider:
If you a 1 meter long stick and a 1.5 meter long stick, isn't the absolute difference between them the same regardless of how you do the length comparison?

Dale
Mentor
U is the voltage between points a and b.

Exactly that's my doubt, are both expressions correct?
They are both correct, but they are not equal to each other. The thing with voltages is that it is a directed difference. Strictly speaking you should never write just voltage $U$. It should always be written more clearly $U_{ab}=-U_{ba}$. You need to always specify which point is being considered to be the reference or the “ground”.

U is the voltage between points a and b
There is no such thing as “the voltage between points a and b”. There is “the voltage at a with respect to b” and “the voltage at b with respect to a”. Often in context which point is used as reference/ground is clear, so it may not always be stated so precisely, but that is the meaning.

Mister T
Gold Member
U is the voltage between points a and b.
That's not a definition because it could refer to $V_b-V_a$ or $V_a-V_b$ or $|{V_b-V_a}|$.

It's like referring to an altitude between two points. For example, if you have an altitude difference of 10 meters between points a and b you don't know if the altitude at point b is 10 meters greater than the altitude at point a or the other way around. Saying that the altitude between them is 10 meters doesn't tell us which of those to choose. And saying that the absolute value of the difference is 10 meters doesn't tell us, either.

Mister T
Gold Member
Consider:
If you a 1 meter long stick and a 1.5 meter long stick, isn't the absolute difference between them the same regardless of how you do the length comparison?
Yes, but knowing only that the difference is 0.5 meters doesn't tell us which of the two items is longer. And it certainly doesn't imply that $1.5-1.0$ is equal to $1.0-1.5$. But that is indeed what the OP's two equations imply about the value of $U$.

Tom.G