2 definitions for voltage, how are they equivalent?

In summary: 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?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##.
  • #1
Granger
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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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  • #2
Not quite sure I understand the question.
Granger said:
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)##
 
  • #3
Doc Al said:
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?
 
  • #4
Granger said:
$$U=V_a - V_b =\int_{a}^{b} \textbf{E} \cdot d\textbf{s}$$

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##.
 
  • #5
Mister T said:
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?
 
  • #6
$$V_a - V_b =\int_{a}^{b} \textbf{E} \cdot d\textbf{s}$$

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##.
 
  • #7
Granger said:
U=Va−Vb=∫baE⋅ds

Granger said:
U=Vb−Va=−∫baE⋅ds

Granger said:
How can both of this expressions represent the potential difference between points A and B?

Granger said:
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?
 
  • #8
Granger said:
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”.

Granger said:
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.
 
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  • #9
Granger said:
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.
 
  • #10
Tom.G said:
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##.
 
  • #11
Mister T said:
doesn't tell us which of the two items is longer.
Agreed.
I also agree that the polarity is often, but not always important. (re: active devices vs resistive heater)

However, the OPs question was:
Granger said:
...definitions of potential difference...
With no reference to polarity. That was the question I was trying to answer.
 

FAQ: 2 definitions for voltage, how are they equivalent?

What is voltage?

Voltage is a measure of the electrical potential difference between two points in an electrical circuit. It is often referred to as the driving force that pushes electricity through a circuit.

How is voltage measured?

Voltage is typically measured in volts (V) using a voltmeter. The voltmeter is connected in parallel to the two points in the circuit for which the potential difference is being measured.

What is the difference between AC and DC voltage?

AC (alternating current) voltage is the type of voltage found in most household outlets and is constantly changing in direction and magnitude. DC (direct current) voltage, on the other hand, flows in only one direction and is typically found in batteries.

What is the relationship between voltage and current?

Voltage and current are directly related to each other in an electrical circuit. The higher the voltage, the greater the potential difference and the higher the current will flow through the circuit.

How are the two definitions of voltage equivalent?

The two definitions of voltage are equivalent because they both describe the same concept of electrical potential difference. Whether it is measured as the amount of work needed to move a unit of charge from one point to another (joules per coulomb) or the amount of energy per unit of charge (volts), they both represent the same physical quantity.

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