Find Potential of Disk of Radius R, Charge Q w/ Equ 1

[Miike012]

I would like to know if there is a way of finding the Potential of a disk of radius R and charge Q with uniform charge distribution using Equ 1 instead of Equ 2?

Equ 1:
ΔV = -∫Edotdx rather than

Equ 2:
ΔV = ∫dU_{elec potential}

For Equ 1 I'm guessing that the equation would be of the form...

ΔV = -∫∫Ecos(∏/2)dA = -∫∫EdA = -∫∫Edxdy
And
Limits of int (X-Direction): -√(R² - y²) to √(R² - y²)
Limits of int (Y-Direction): -√(R² - x²) to √(R² - x²)

And E would be the EField of a disk...

 

[vela]

You can certainly integrate the E field to find the potential, but your attempt is wrong. It looks like you're confusing calculating the potential with calculating E.

To use equation 1, you need to have already found $\vec{E}$, and you'd calculate the line integral along any path C that runs from infinity to the point $\vec{r}$:
$$ V(\vec{r}) = -\int_C \vec{E}\cdot d\vec{x}.$$ To find $\vec{E}$, you'd calculate the contribution from each piece of the disk to the electric field at some point in space, and then integrate over the entire disk.

To use equation 2, you take a similar approach as you do when calculating $\vec{E}$. You calculate the contribution from each piece of the disk to the electric potential at some point in space, and then integrate over the entire disk. Calculating electric potential is a bit easier than calculating the field because the potential is a scalar quantity. You don't have to worry about summing vectors like you do when calculating the electric field.

 

[Miike012]

You can certainly integrate the E field to find the potential, but your attempt is wrong. It looks like you're confusing calculating the potential with calculating E.

To use equation 1, you need to have already found $\vec{E}$, and you'd calculate the line integral along any path C that runs from infinity to the point $\vec{r}$:
$$ V(\vec{r}) = -\int_C \vec{E}\cdot d\vec{x}.$$ To find $\vec{E}$, you'd calculate the contribution from each piece of the disk to the electric field at some point in space, and then integrate over the entire disk.

So equation 1 would be..

V = -∫E_diskdx and integrate from x_f = distance from point to disk and x_i = ∞. is that correct?

 

[vela]

Yes, basically.

 

[Nickie]

Yes, it is possible to use Equation 1 to find the potential of a disk with uniform charge distribution. Equation 1 is known as the line integral form of the electric potential, while Equation 2 is known as the potential energy form. Both equations are equivalent and can be used interchangeably.

To use Equation 1, you would need to first calculate the electric field of the disk at a given point. This can be done using Gauss's law or by integrating the electric field equation for a charged disk. Once you have the electric field, you can then integrate it along a path from a reference point to the point where you want to find the potential. This path can be chosen in any way as long as it ends at the desired point.

In the case of a disk with uniform charge distribution, the electric field will be constant at any point on the disk's surface. Therefore, the path of integration can be chosen to simply go from the center of the disk to the desired point. The limits of integration will depend on the geometry of the disk and can be calculated using the equations you have provided.

Overall, both Equation 1 and Equation 2 can be used to find the potential of a disk with uniform charge distribution. It is up to the researcher to choose which equation is more convenient to use in a given situation.