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For example, my textbook clearly says that the Potencial Difference between points A and B is given by $$ V_{AB} \equiv V_A-V_B = \int_A^B \vec E\cdot d\vec l $$

but I've seen, in other textbooks, the Potencial Difference between points A and B defined like this $$ \Delta V \equiv V_B-V_A = -\int_A^B \vec E\cdot d\vec l $$

Moreover, both of these definitions state that the test charge is being moved from A to B in an electric field.

So, is the Potencial Difference between points A and B $V_A-V_B $ or is it $ V_B-V_A $ ?

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  • $\begingroup$ How are $V_{AB}$ and $\Delta V$ defined (i.e. in a diagram showing what nodes are being referred to)? $\endgroup$
    – The Photon
    Commented Mar 9, 2020 at 20:19
  • $\begingroup$ There's no nodes. The first definition is from the book: Introductory Electromagnetics (Prentice-Hall, Inc.); Zoya Popovic, Branco D. Popovic . $\endgroup$
    – PTSONIC
    Commented Mar 9, 2020 at 20:33
  • $\begingroup$ The second one is from the book: Physics for Scientists and Engineers (Cengage Learning); Raymond A. Serway, John W. Jewett $\endgroup$
    – PTSONIC
    Commented Mar 9, 2020 at 20:34
  • $\begingroup$ Please edit your question to share enough context. Not everybody has access to those particular books. $\endgroup$
    – The Photon
    Commented Mar 9, 2020 at 20:55

2 Answers 2

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Because it is a potential difference, it doesn't matter.

For example: "I'm 10cm taller than my dad" and "my dad is 10cm smaller than me" contain the same information, just expressed differently.

Similarly, suppose I measure $V_{AB}$ and say, "$V_{AB}$ is 5V", and you say "actually it is -5V", we still agree on the actual quantity, but just disagree on how we should express it.

The only caveat here is that when you perform a calculation, you must pick a convection (either one) and stick to it, but there is normally a 'natural' one for the problem at hand.

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  • $\begingroup$ I thought 5V and -5V meant two different things as I was told that if $V_{AB} >0 $ then A was at an higher potential than B (therefore, field lines go from A to B) and if $V_{AB} <0 $ then B was at an higher potential than A (therefore, field lines go from B to A). $\endgroup$
    – PTSONIC
    Commented Mar 9, 2020 at 20:48
  • $\begingroup$ Electric field lines point down the voltage gradient. Deciding if $V_{AB}$ is $\pm 5$V amounts to deciding which way your spatial axis points (this is what I mean about sticking to a convention) which ensures you get the same answer for the electric field. $\endgroup$
    – jobla6
    Commented Mar 9, 2020 at 22:14
  • $\begingroup$ For example, suppose I have a 5V battery connected to a 1$\Omega$ resistor. Going out of the positive terminal of the battery, I measure a 5V potential on one side of the resistor and 0V on the other, giving a potential difference of 5V so we get a 5A current flowing out of the positive terminal. Equally, we could leave the negative terminal of the cell and measure 0V and then 5V at the resistor, giving a potential difference of -5V. This causes a -5A current to leave the negative terminal, which is equivalent to a 5A current leaving the positive terminal. $\endgroup$
    – jobla6
    Commented Mar 9, 2020 at 22:19
  • $\begingroup$ Thank you, great explanation! $\endgroup$
    – PTSONIC
    Commented Mar 10, 2020 at 2:00
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The physics of both equations is the same. Look at the second equality in each one. Both agree on $V_B-V_A$. It’s only ambiguous notations $V_{AB}$ and $\Delta V$ that are confusing.

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  • $\begingroup$ They're the notations used by each book but they both refer it as Potencial Difference $\endgroup$
    – PTSONIC
    Commented Mar 9, 2020 at 20:36
  • $\begingroup$ $V_B-V_A$ and $V_A-V_B$ are both valid ways to express the concept of “potential difference between two points”. There is not one correct way. It’s best to focus on the physics, not on the terminology. The physics is that the field is directed from high potential to low potential. $\endgroup$
    – G. Smith
    Commented Mar 9, 2020 at 22:00

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