Mathematics of the Faraday cage
The amplitude of the gradient of a potential inside a wire cage is investigated, with particular attention to the 2D configuration of a ring of n disks of radius r held at equal potential. The Faraday shielding effect depends upon the wires having finite radius and is weaker than one might expect, s...
| Autores principales: | , , |
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| Formato: | Journal article |
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Society for Industrial and Applied Mathematics
2015
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| _version_ | 1826291084196577280 |
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| author | Hewett, D Trefethen, L Chapman, S |
| author2 | Pavliotis, G |
| author_facet | Pavliotis, G Hewett, D Trefethen, L Chapman, S |
| author_sort | Hewett, D |
| collection | OXFORD |
| description | The amplitude of the gradient of a potential inside a wire cage is investigated, with particular attention to the 2D configuration of a ring of n disks of radius r held at equal potential. The Faraday shielding effect depends upon the wires having finite radius and is weaker than one might expect, scaling as |log r|/n in an appropriate regime of small r and large n. Both numerical results and a mathematical theorem are provided. By the method of multiple scales, a continuum approximation is then derived in the form of a homogenized boundary condition for the Laplace equation along a curve. The homogenized equation reveals that in a Faraday cage, charge moves so as to somewhat cancel an external field, but not enough for the cancellation to be fully effective. Physically, the effect is one of electrostatic induction in a surface of limited capacitance. An alternative discrete model of the effect is also derived based on a principle of energy minimization. Extensions to electromagnetic waves and 3D geometries are mentioned. |
| first_indexed | 2024-03-07T02:54:05Z |
| format | Journal article |
| id | oxford-uuid:aeacd580-d66f-407d-8a2f-b0c74e41a2a0 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:54:05Z |
| publishDate | 2015 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:aeacd580-d66f-407d-8a2f-b0c74e41a2a02022-03-27T03:44:11ZMathematics of the Faraday cageJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:aeacd580-d66f-407d-8a2f-b0c74e41a2a0Symplectic Elements at OxfordSociety for Industrial and Applied Mathematics2015Hewett, DTrefethen, LChapman, SPavliotis, GThe amplitude of the gradient of a potential inside a wire cage is investigated, with particular attention to the 2D configuration of a ring of n disks of radius r held at equal potential. The Faraday shielding effect depends upon the wires having finite radius and is weaker than one might expect, scaling as |log r|/n in an appropriate regime of small r and large n. Both numerical results and a mathematical theorem are provided. By the method of multiple scales, a continuum approximation is then derived in the form of a homogenized boundary condition for the Laplace equation along a curve. The homogenized equation reveals that in a Faraday cage, charge moves so as to somewhat cancel an external field, but not enough for the cancellation to be fully effective. Physically, the effect is one of electrostatic induction in a surface of limited capacitance. An alternative discrete model of the effect is also derived based on a principle of energy minimization. Extensions to electromagnetic waves and 3D geometries are mentioned. |
| spellingShingle | Hewett, D Trefethen, L Chapman, S Mathematics of the Faraday cage |
| title | Mathematics of the Faraday cage |
| title_full | Mathematics of the Faraday cage |
| title_fullStr | Mathematics of the Faraday cage |
| title_full_unstemmed | Mathematics of the Faraday cage |
| title_short | Mathematics of the Faraday cage |
| title_sort | mathematics of the faraday cage |
| work_keys_str_mv | AT hewettd mathematicsofthefaradaycage AT trefethenl mathematicsofthefaradaycage AT chapmans mathematicsofthefaradaycage |