Path orientation for calculating electric potential

In summary: So it is correct to say that the path is from a to b.In summary, the conversation discusses the direction of the path in line integrals in vector calculus. The speaker mentions that they usually see the path oriented from a to b, but their textbook shows it in the opposite direction. They question how to determine the direction of dl and the importance of the starting point of the integration path. The expert responds by explaining that the direction of dl does not matter as long as the starting point is not at the singularity. They also mention that the direction of the path can be reversed without affecting the result.
  • #1
OmegaKV
22
1
For line integrals in vector calculus,

[tex]\int^a_b F \cdot dl[/tex]

I almost always see the path oriented from a to b.

But my textbook has the following (look at the first equation for V(r):

lapJZ5b.jpg


Since the integral's limits are from O to r, I would have expected dl to also be pointing in the direction from O to r (i.e. pointing in the radially inward (minus r hat) direction), but the math in the textbook seems to imply that dl points radially outward (positive r hat direction, from r to O). I say this because the result of E dot dl has no minus sign in front of it.

How do you know which direction to orient dl?
 
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  • #2
It doesn't matter at which point you start your integration path as long as it is not at the singularity at ##\vec{r}=0## of the Coulomb potential (I don't know what's discussed in your book, because you didn't tell us; maybe it's a spherical surface carrying some charge?). Changing the starting point of your path just adds a constant to the potential, but the only thing interesting is the field ##\vec{E}=-\vec{\nabla} V##.

Here they used a path starting from ##r=|\vec{r}| \rightarrow \infty##, making the potential ##0## at infinity, which is a convenient standard choice. Since the field is radial always, the only contribution is from the part going from infinity radially in, and you get the integral solved in your book.
 
  • #3
OmegaKV said:
For line integrals in vector calculus,

[tex]\int^a_b F \cdot dl[/tex]

I almost always see the path oriented from a to b.

But my textbook has the following (look at the first equation for V(r):

lapJZ5b.jpg


Since the integral's limits are from O to r, I would have expected dl to also be pointing in the direction from O to r (i.e. pointing in the radially inward (minus r hat) direction), but the math in the textbook seems to imply that dl points radially outward (positive r hat direction, from r to O). I say this because the result of E dot dl has no minus sign in front of it.

How do you know which direction to orient dl?

The path goes from b to a. If you reverse the order the modulus stay the same but the sign changes.
 

1. What is path orientation for calculating electric potential?

Path orientation for calculating electric potential is a method used in physics to determine the electric potential at a specific point. It involves choosing a path or route from a reference point to the desired point and calculating the potential difference along that path.

2. How is path orientation related to electric potential?

Path orientation is directly related to electric potential because it allows us to calculate the potential difference between two points. This potential difference, also known as voltage, is a measure of the electric potential at those points.

3. Why is path orientation important in physics?

Path orientation is important in physics because it helps us understand the behavior of electric fields and how they affect the movement of charged particles. It also allows us to calculate the work done by the electric field on a charged particle.

4. How do you determine the direction of the path for calculating electric potential?

The direction of the path for calculating electric potential is determined by the direction of the electric field. The path should always be oriented in the direction opposite to the electric field lines.

5. Can path orientation be used for any type of electric field?

Yes, path orientation can be used for any type of electric field, whether it is uniform, non-uniform, or a combination of both. It is a versatile method that can be applied to various situations in physics.

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