Understanding the concept of voltage

In summary, the concept of voltage is a difference in electric potential between two points, defined as the difference between the potentials at those points. The notation of ##V_{ab}## and ##\Delta{V}## can be used interchangeably, with the latter being a difference between a final and initial position. The symbols ##a## and ##b## represent extrema of integration, with the integration path being arbitrary. The integrand, ##dl##, represents an arbitrary curve connecting a fixed point with a variable point, and the direction can be switched by switching the lead wires of a voltmeter.
  • #1
luca54
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Hi everyone!

I ask some help in understanding better the concept of voltage. The voltage is a difference in electric potential between two points ##a## and ##b##. It is defined as

1579042968372.png


However, I'm a bit confused with the use of notation:

- Is ##V_{ab}## the same as ##\Delta{V}##, or rather ##-\Delta{V}##? In fact, ##V_{ab}## is also written as ##V_a-V_b##, while ##\Delta{V}## should be a difference between a final and an initial position.
- What do ##a## and ##b## represent? They are extrema of integration, but how do we select them in a problem, one as the starting position and the other as the arrival? What does the integration from one to the other (and not vice versa) mean?

Eventually, I would like to add another question, this one about the integrand:

- What is concretely ##dl##, and what is its direction?

Thanks very much!
 

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  • #2
Looks like you are copy-pasting from stack exchange. For inline LaTeX you have to replace the SE $ with the PF ##

I reverted and updated for you
 
  • #3
Ah thanks a lot!
Yes, I've posted the question also there, but it hasn't been directly answered, and my doubts are still there :confused::frown:
 
  • #5
This is only true for electrostatics, because only then the electric field has a potential, independent of the integration path in your formula, i.e., you can use any path connecting the points with the position vectors ##\vec{x}_a## and ##\vec{x}_b##. The potential is defined as
$$V(\vec{x})=-\int_{C(\vec{x}_0,\vec{x})} \mathrm{d} \vec{r} \cdot \vec{E}(\vec{r}),$$
where ##C(\vec{x}_0,\vec{x})## is an arbitrary curve connecting an arbitrary fixed point ##\vec{x}_0## with the variable point ##\vec{x}##. Then you have
$$\vec{E}(\vec{x})=-\vec{\nabla} V(\vec{x}).$$
The voltage is simply the difference of the potentials between the two points in question,
$$\Delta V=V(\vec{x}_b)-V(\vec{x}_a).$$
Since the line integral defining ##V## only depends on the initial an final points of the path, you get
$$\Delta V=-\int_{C'(\vec{x}_a,\vec{x}_b)} \mathrm{d} \vec{r} \cdot \vec{E}(\vec{r}),$$
where ##C'(\vec{x}_a,\vec{x}_b)## is an arbitrary path connecting the points at ##\vec{x}_a## and ##\vec{x}_b##.
 
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  • #7
vanhees71 said:
This is only true for electrostatics, because only then the electric field has a potential, independent of the integration path in your formula, i.e., you can use any path connecting the points with the position vectors ##\vec{x}_a## and ##\vec{x}_b##. The potential is defined as
$$V(\vec{x})=-\int_{C(\vec{x}_0,\vec{x})} \mathrm{d} \vec{r} \cdot \vec{E}(\vec{r}),$$
where ##C(\vec{x}_0,\vec{x})## is an arbitrary curve connecting an arbitrary fixed point ##\vec{x}_0## with the variable point ##\vec{x}##. Then you have
$$\vec{E}(\vec{x})=-\vec{\nabla} V(\vec{x}).$$
The voltage is simply the difference of the potentials between the two points in question,
$$\Delta V=V(\vec{x}_b)-V(\vec{x}_a).$$
Since the line integral defining ##V## only depends on the initial an final points of the path, you get
$$\Delta V=-\int_{C'(\vec{x}_a,\vec{x}_b)} \mathrm{d} \vec{r} \cdot \vec{E}(\vec{r}),$$
where ##C'(\vec{x}_a,\vec{x}_b)## is an arbitrary path connecting the points at ##\vec{x}_a## and ##\vec{x}_b##.

Thanks very much for the answer!
 
  • #8
luca54 said:
What does the integration from one to the other (and not vice versa) mean?
Since you have responses about the other portions, I thought I would address this. If you have a typical voltmeter then A will be your red lead wire and B will be your black lead wire. So integrating from A to B or from B to A is just a matter of switching your lead wires.
 
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FAQ: Understanding the concept of voltage

1. What is voltage?

Voltage is a measure of the electric potential difference between two points in an electric circuit. It is often described as the force that pushes electric charges through a conductor.

2. How is voltage measured?

Voltage is measured in volts (V) using a voltmeter. It is typically measured between two points in a circuit, such as the positive and negative terminals of a battery.

3. What is the relationship between voltage and current?

Voltage and current are directly proportional, meaning that an increase in voltage will result in an increase in current, and vice versa. This relationship is described by Ohm's Law: V = IR, where V is voltage, I is current, and R is resistance.

4. What is the difference between AC and DC voltage?

AC (alternating current) voltage constantly changes direction, while DC (direct current) voltage flows in one direction. AC voltage is used in most household circuits, while DC voltage is commonly used in electronic devices and batteries.

5. How does voltage affect the functioning of electronic devices?

Voltage is essential for the functioning of electronic devices as it powers the flow of electric current through the circuit. The correct voltage is necessary to ensure that the device operates properly and does not get damaged.

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