en>BD2412: dab fix

2014-05-23T13:38:29Z

dab fix

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	~~The mutual inductance by circuit ''i'' on circuit ''j''~~ is ~~given by~~ the ~~double integral~~ ''[[~~Franz Ernst Neumann\|Neumann]] formula''~~		Alyson Meagher is the name her mothers and fathers gave her but she doesn't like when individuals use her complete name. She is really fond of caving but she doesn't have the time recently. I am currently a psychic chat online [[http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going www.zavodpm.ru]] travel agent. For a while I've been in Mississippi but now I'm contemplating other options.<br><br>My blog: free [http://modenpeople.co.kr/modn/qna/292291 best psychic readings] reading ([http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc gcjcteam.org])
	:~~<math> M_{ij} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j} \frac{\mathbf{ds}_i\cdot\mathbf{ds}_j}{\|\mathbf{R}_{ij}\|} <~~/~~math>~~

	~~==Derivation ==~~
	~~:<math> \Phi_{i} = \int_{S_i} \mathbf{B}\cdot\mathbf{da} = \int_{S_i} (\nabla\times\mathbf{A})\cdot\mathbf{da}~~
	~~= \oint_{C_i} \mathbf{A}\cdot\mathbf{ds} = \oint_{C_i} \left(\sum_{j}\frac{\mu_0 I_j}{4\pi} \oint_{C_j} \frac{\mathbf{ds}_j}{\|\mathbf{R}\|}\right) \cdot \mathbf{ds}_i <~~/~~math>~~
	~~where~~

	~~:<math>\Phi_i\ \,<~~/~~math> is the [[magnetic flux]] through the ''i''th surface by the [[electrical circuit]] outlined by ''C''<sub>''j''<~~/~~sub>~~
	~~:''C<sub>i<~~/~~sub>'' is the enclosing curve of S<sub>''i''</sub>~~.
	~~:''B'' is the [[magnetic field]] vector~~.
	~~:''A'' is the [[vector potential~~]]. ~~<ref>{{cite book \|last=Jackson \|first=J. D. \|title=Classical Electrodynamics \|url= \|date=1975 \|accessdate= \|edition= \|publisher=Wiley \|pages=176, 263 \|isbn= }}</ref>~~

	~~[[Stokes~~' ~~theorem]] has~~ been ~~used.~~

	~~:<math> M_{ij} \ \stackrel{\mathrm{def}}{=}\ \frac{\Phi_{i}}{I_j} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j} \frac{\mathbf{ds}_i\cdot\mathbf{ds}_j}{\|\mathbf{R}_{ij}\|} </math>~~
	~~so that the mutual inductance is a purely geometrical quantity independent of the current~~ in ~~the circuits.~~

	~~==Self inductance ==~~
	~~In the self inductance case ''C<sub>i</sub>=C<sub>j</sub>'~~'. Therefore ''1/R'' diverges and the finite radius of the wire and the distribution of the current in the wire must be taken into account. A generic formula for the self inductance ''M'' of a wire loop is available provided that the length ''l'' of the wire is much larger than its radius ''a'',<~~ref name="rden12"~~>~~{{cite arXiv \|last=Dengler \|first=R. \|eprint=1204.1486 \|title=Self inductance of a wire loop as a curve integral\|year=2012 }}~~<~~/ref~~>

	:~~<math> M = M_{ii} = \frac{\mu_0}{4\pi} \left ( \oint_{C}\oint_{C'} \frac{\mathbf{ds}\cdot\mathbf{ds}'}{\|\mathbf{R}_{ss^{\prime }}\|}\right )_{\|\mathbf{R}\| > a/2}~~
	~~+ \frac{\mu_0}{2\pi}lY + O\left( \mu_0 a \right ).<~~/~~math>~~
	~~Points with ''\|R\| < a~~/~~2'' now must be excluded from the curve integral~~. ~~The correction term proportional to ''l'' originates from short wire segments which are essentially cylinders. ''Y=0'' if the current flows in the surface of the~~
	~~wire, ''Y=1/4'' if the current is homogeneous in the wire~~.

	~~The error of the formula is of order ''μ<sub>0<~~/~~sub>a'' if the current loop contains sharp corners and of~~
	~~order ''μ<sub>0<~~/~~sub>a<sup>2<~~/~~sup>/R<sub>c<~~/~~sub>'' for smooth current loops with minimal curvature radius ''R<sub>c<~~/~~sub>''~~.~~<ref name="rden12"~~/>

	~~In the skin effect case another approximation is hidden in the assumption of constant current density.~~
	~~If wires are close to each other additional currents flow in the surface of the wires (expelling the magnetic field).~~
	~~In this case Maxwell's equation must be solved to determine currents and fields~~.

	==~~References~~==
	~~{{reflist}}~~

	~~[[Category:Electrodynamics]~~]

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