Interaction energy between two identical hemispherical surfaces with uniform surface charge density

Ciftja, Brent; Colbert-Pollack, Cal; Ciftja, Orion; Littlejohn, Lindsey

doi:10.1088/1361-6404/ac2b05

Cited by 3 publications

(5 citation statements)

References 30 publications

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“…The result above can be easily verified if one starts from the expression in equation (20) and applies the following integral formula:…”

Section: Axis Of Annulusmentioning

confidence: 81%

“…In many instances, integrals over angular variables of two-particle functions are very difficult [17][18][19][20]. However, one can see that axial symmetry in conjunction with equation (15) helps a lot in this case to obtain: Electrostatic potential on the plane of a uniformly charged disk, V Disk (ρ, z = 0, R) (filled circles) and a uniformly charged ring, V Ring (ρ, z = 0, R) (solid line) as a function of distance ρ/R.…”

Section:  mentioning

confidence: 99%

“…To simplify the calculations, one introduces a dummy variable, q = k R 2 and after some careful manipulation of the terms in equation (20), the final result reads:…”

Section: ( ) ¹mentioning

confidence: 99%

“…The 1D integrals can be easily calculated numerically with very high precision by using standard integration packages [21]. At this stage, one can use either equation (20) or (25) to verify whether the above quantities reduce to known results for special cases. To be more specific, we will look at the disk and ring limit of the electrostatic potential of the uniformly charged annulus, value of electrostatic potential at center of annulus, value of electrostatic potential on the plane of the annulus and value of electrostatic potential along the axis of symmetry of the annulus.…”

Section: ( ) ¹mentioning

confidence: 99%

“…A uniformly charged annulus becomes a uniformly charged disk with radius, R 2 when R 1 = 0. Therefore, one sets R 1 = 0 in equation (20), to obtain:…”

Section: Disk Limitmentioning

confidence: 99%

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Electrostatic potential of a uniformly charged annulus

Ciftja,

Bentley Jr

2024

Eur. J. Phys.

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The calculation of the electrostatic potential and/or electrostatic ﬁeld due to a continuous distribution of charge is a well-covered topic in all calculus-based undergraduate physics courses. The most common approach is to consider bodies with uniform charge distribution and obtain the quantity of interest by integrating over the contributions from all the diﬀerential charges. The examples of a uniformly charged disk and ring are prominent in many physics textbooks since they illustrate well this technique at least for special points or directions of symmetry where the calculations are relatively simple. Surprisingly, the case of a uniformly charged annulus, namely, an annular disk, is largely absent from the literature. One might speculate that a uniformly charged annulus is not extremely interesting since after all, it is a uniformly charged disk with a central hole. However, we show in this work that the electrostatic potential created by a uniformly charged annulus has features that are much more interesting than one might have expected. A uniformly charged annulus interpolates between a uniformly charged disk and ring. However, the results of this work suggest that a uniformly charged annulus has such electrostatic features that may be essentially viewed as solely ring-like. The ring-like characteristics of the electrostatic potential of a uniformly charged annulus are evident as soon as a hole is present no matter how small the hole might be. The solution of this problem allows us to draw attention to the pedagogical aspects of this overlooked, but very interesting case study in electrostatics. In our opinion, the problem of a uniformly charged annulus and its electrostatic properties deserves to be treated at more depth in all calculus-based undergraduate physics courses covering electricity and magnetism.

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“…The result above can be easily verified if one starts from the expression in equation (20) and applies the following integral formula:…”

Section: Axis Of Annulusmentioning

confidence: 81%

Section:  mentioning

confidence: 99%

“…To simplify the calculations, one introduces a dummy variable, q = k R 2 and after some careful manipulation of the terms in equation (20), the final result reads:…”

Section: ( ) ¹mentioning

confidence: 99%

Section: ( ) ¹mentioning

confidence: 99%

“…A uniformly charged annulus becomes a uniformly charged disk with radius, R 2 when R 1 = 0. Therefore, one sets R 1 = 0 in equation (20), to obtain:…”

Section: Disk Limitmentioning

confidence: 99%

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Electrostatic potential of a uniformly charged annulus

Ciftja,

Bentley Jr

2024

Eur. J. Phys.

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Coulomb self-energy of a solid hemisphere with uniform volume charge density

Ciftja

2023

Mod. Phys. Lett. B

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Calculation of the Coulomb self-energy of a solid hemisphere with uniform volume charge density represents a very challenging task. This system is an interesting example of a body that lacks spherical symmetry though it can be conveniently dealt with in spherical coordinates. In this work, we explain how to calculate the Coulomb self-energy of a solid hemisphere with uniform volume charge density by using a method that relies on the expansion of the Coulomb potential as an infinite series in terms of Legendre polynomials. The final result for the Coulomb self-energy of a uniformly charged solid hemisphere turns out to be quite simple.

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Calculation of the Coulomb Self-Energy of a Spherical Surface with Uniform Surface Charge Density Using the Fourier Transform Method

Ciftja

2022

The Phys. Educat.

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Undergraduate students in upper levels of physics or engineering programs learn the theory of Fourier series and integral transform method from mathematics courses. Nevertheless, they rarely see the application of such a method to solving problems in calculus-based physics courses that deal with topics such as electrostatics or magnetism. In this work, we illustrate the utility of the Fourier transform method by considering and solving via such a technique a representative problem that arises in electrostatics. The chosen case study is that of a spherical surface with uniform surface charge density and the calculation of its electrostatic Coulomb self-energy. By solving this problem by using the Fourier transform technique we also draw attention to the pedagogical aspects of the treatment. In particular, we stress the point that the Fourier transform method should be treated at more depth in calculus-based physics courses for undergraduate students.

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Interaction energy between two identical hemispherical surfaces with uniform surface charge density

Cited by 3 publications

References 30 publications

Electrostatic potential of a uniformly charged annulus

Electrostatic potential of a uniformly charged annulus

Coulomb self-energy of a solid hemisphere with uniform volume charge density

Calculation of the Coulomb Self-Energy of a Spherical Surface with Uniform Surface Charge Density Using the Fourier Transform Method

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