Teorema Gauss

Amsa da cewa akwai jirgin karamin karami ko karamin kudasa a kan wani abu mai karfi ko karamin kudasa, kuma a cikin wannan jirgi na karamin karami akwai hanyoyi masu kyauko flux. Wannan flux na fitar da karamin karami. Yanzu adadin wannan hanyoyin flux ya shafi adadin karamin karami daga yake ita ce. Don samun wannan muryar, an gina Gauss’s theorem. Wannan theorem ya zama daya daga cikin manyan teoremin da suka fi shahara a fagen ilimin karamin karami. A tunanen wannan theorem za a iya samun adadin flux da ya fitar da duk fadadduka mai karfi.

Wannan theorem ta ce a cikin kowane fadada mai karfi da ke jagoranci abu mai karfi, adadin electric flux ya zama da take da karamin karami mai karfi a kan fadaddukan.
Idan ake amfani da karamin karami Q1, Q2_ _ _ _Qi, _ _ _ Qn da ke jagoranci fadaddukan, inda wannan theorem zai iya bayyana da tushen integral:

Inda, D shine flux density a coulombs/m2 kuma dS shine vector mai sani.

Bayani a kan Gauss’s Theorem

Don bayyana Gauss’s theorem, ya kamata a taka misali don samun fahimta daidai.
Idan Q shine karamin karami a kan tsohon kugurbin, kuma flux da ya fitar da karamin karami ya zama da tsaye a kan fadaddukan. Wannan theorem ta ce a cikin wannan misali, adadin flux da ya fitar da karamin karami ya zama Q coulombs, kuma wannan za a iya tabbatar da shi da tarihi. Amma idan karamin karami ba a kan tsohon kugurbin bane a kan wurin (kamar hakan a kan taswiran).

A wannan lokacin, lissafin flux suna da tsaye a kan fadaddukan mai karfi, kuma a nan suna saude da biyu masu kyau, wanda ya zama sinθ component da cosθ component. Idan ake sume waɗannan components daga dukkan karamin karami, adadin net result ya zama da karamin karami mai karfi, wanda ya tabbatar da Gauss’s theorem.

Dalilin Gauss’s Theorem

Za a duba abu mai karfi Q a kan medium mai karfi mai karfi da permittivity ε.

Electric field intensity a kan kowane wurin a kan adadin r daga karamin karami ya zama

Flux density ya zama,

Daga taswiran, flux a kan area dS

Inda, θ shine angle bayan D da normal to dS.
Daga bayanin solid angle

Inda, dΩ shine solid angle da ke jagoranci Q a kan elementary surface are dS. Saboda haka, adadin displacement na flux a kan duk fadaddukan ya zama

Saboda a mince, solid angle da ke jagoranci kowane fadaddukan ya zama 4π steradians, saboda haka, adadin electric flux a kan duk fadaddukan ya zama

Wannan shine form na integral na Gauss’s theorem. Saboda haka, wannan theorem ya tabbatar da shi.

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