Uniformly-charged disk electric field at ANY point (i.e., off-axis)?

In summary, the conversation discusses finding the equation for the electric field intensity of a uniformly charged disk in the xy-plane with radius and surface charge density, and the difficulties in evaluating the integral required for the solution. Suggestions are made to evaluate the potential first and then use Legendre polynomials for the expansion of the integrand. It is noted that the solution may require knowledge outside of the course.
  • #1
TTTTTTrent
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Homework Statement



I need to find the equation for the electric field intensity [tex]\vec{E}(\vec{R})[/tex] of a uniformly-charged disk lying in the xy-plane with radius [tex]\rho_{0}[/tex] and surface charge density [tex]\rho_{s}'[/tex] at any point.

Homework Equations



I know that a charged disk doesn't have the symmetry needed for Gauss's Law to be useful, and finding the electric field intensity on the z axis is easily found using Coulomb's Law, so I'm using that again for this more general problem.

The Attempt at a Solution



I set up the position vectors for the observation point and the source point, with

[tex]\vec{R} = \rho\hat{\rho}+z\hat{z}[/tex] for the observation point
[tex]\vec{R}' = \rho'\hat{\rho}'[/tex] for the source point (prime designating source quantities throughout this problem).

Using cylindrical coordinates I found dA to be [tex]\rho'd\phi'd\rho'[/tex]. When plugging the position vectors into the integral, though, I had to convert [tex]\hat{\rho}[/tex] and [tex]\hat{\rho}'[/tex] to x and y components because they didn't correspond to the same unit vectors and also varied with [tex]\phi '[/tex]. Taking the constant surface charge density out of the equation and simplifying as best I could with the magnitude in the denominator, I was left with the integral

[tex]\vec{E}(\vec{R}) = \frac{\rho_{s}'}{4\pi\epsilon_{0}}\int^{\rho_{0}}_{\rho '=0}\rho'd\rho'\int^{2\pi}_{\phi '=0}\frac{(\rho cos \phi - \rho'cos \phi')\hat{x} + (\rho sin \phi - \rho'sin \phi')\hat{y} + z\hat{z}}{[\rho^{2}-2\rho\rho'cos(\phi - \phi') + (\rho')^{2} + z^{2}]^{3/2}}d\phi'[/tex]

As you can see, this integral seems like it would take some monumental calculus skills to evaluate. I've tried breaking the integral up into fractions with numerators being each term of the original numerator, and a common denominator for all. Even with the [tex]z\hat{z}[/tex] term I can't see a way to evaluate that integral. There seem to be too many nested functions within one another. I've looked around online and there are thousands of pages solving an on-axis charged disk problem, but I've only found 2 or 3 sites that even discuss solving an off-axis one. They've referred to elliptic integrals and legendre polynomials...both of which I know nothing about and which haven't been spoken of in my course ever. Am I missing something? I've never been the strongest in integration techniques, but this is just ridiculous! Help please?!
 
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  • #2
I suggest evaluating the potential first and then obtain the field by taking a derivative.
Even so, it is very unlikely that you will be able to get the solution in a closed form. It seems you should expand the integrand in terms of Legendre polynomials. Consult with Jackson's EM book or hopefully, Wiki.
 
  • #3
Thanks for the quick reply and good suggestions, weejee. My professor likes to watch us squirm, so I'm not surprised that the solution to this problem will probably require knowledge outside of our course. I'll definitely try the potential to field method first, though. Good thinking.
 

FAQ: Uniformly-charged disk electric field at ANY point (i.e., off-axis)?

1. What is a uniformly-charged disk electric field?

A uniformly-charged disk electric field is an electric field that is created by a disk-shaped object with a constant and evenly-distributed charge. This type of electric field is commonly used in physics and engineering to model the behavior of charged disks.

2. How is the electric field calculated at any point in space?

To calculate the electric field at any point in space, the formula E = kQ/r^3 is used, where E is the electric field strength, k is the Coulomb's constant, Q is the total charge of the disk, and r is the distance from the disk to the point of interest.

3. How does the electric field vary off-axis?

The electric field varies off-axis in a non-linear manner, meaning that it does not follow a straight line. As the distance from the center of the disk increases, the electric field strength decreases. This decrease is steeper at points closer to the edge of the disk compared to points closer to the center.

4. How does the position of the point affect the electric field strength?

The position of the point with respect to the disk affects the electric field strength. Points that are closer to the disk will experience a stronger electric field compared to points that are farther away. Additionally, points that are closer to the edge of the disk will experience a stronger electric field compared to points that are closer to the center.

5. Can the electric field at any point off-axis be negative?

Yes, the electric field at any point off-axis can be negative. This indicates that the direction of the electric field is pointing in the opposite direction compared to the direction of the electric field at points on the opposite side of the disk. The magnitude of the electric field, however, will always be positive.

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