Lankide:Ingeniaritza-gradua/Faradayren paradoxa

Michael Faraday

Faraday-ren paradoxa da edozein esperimentu non Michael Faraday-ren Indukzio elektromagnetikoko legeak okerreko emaitzak iragartzen bide dituen. Paradoxa hauek bi motatakoak izan daitezke:

• Faraday-ren legeak zero IEE (indar elektroeragile) iragartzen du egiatan IEE zero ez denean.
• Faraday-ren legeak IEE ez nulua iragartzen du errealitatean IEE zero denean.

Faradayk bere indukzio legea 1831. urtean ondorioztatu zuen, lehen sorgailu elektromagnetikoa edo dinamoa asmatu ondoren, baina ez zegoen batere pozik berak emandako paradoxaren azalpenarekin.

Faraday-ren legea Maxwell-Faraday ekuazioarekin alderatuta

Faraday-ren legeak ( Faraday-Lenz legea izenarekin ere ezaguna dena) adierazten du indar elektroeragilea (IEE) fluxu magnetikoaren t denborarekiko deribatu totala dela:

Interpretazio errorea (Errore sintaktikoa): {\displaystyle <mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script"> <math /> </mi></mrow></mrow><mo> <math /> </mo><mo> <math /> </mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi> <math /> </mi><msub><mi mathvariant="normal"> <math /> </mi><mrow class="MJX-TeXAtom-ORD"><mi> <math /> </mi></mrow></msub></mrow><mrow><mi> <math /> </mi><mi> <math /> </mi></mrow></mfrac></mrow><mo> <math /> </mo></mrow> } ${\displaystyle }$ ${\displaystyle }$ </img>

non Interpretazio errorea (MathML posible bada (proba fasean): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/eu.wikipedia.org/v1/":): {\displaystyle <mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script"> <math /> </mi></mrow></mrow></mrow> } ${\displaystyle }$ ${\displaystyle }$ </img> IEE eta Φ _B fluxu magnetikoa diren. Indar elektroeragilearen noranzkoa Lenzen legeak ematen du. Izan ere, sarritan ahaztu egiten da Faraday-ren legea fluxu magnetikoaren deribatu osoan oinarritzen dela eta ez deribatu partzialean. Horrek esan nahi du IEE bat sor daitekeela gainazalaren zeharreko fluxu osoa konstantea bada ere. Arazo hau gainditzeko teknika bereziak erabil daitezke. Ikusi beherago Faraday-ren legearekin erabil daitezkeen teknika bereziei buruzko atala. Halere, Faradayren legeko interpretaziorik ohikoena honako hau da:

«

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit.

»

Faraday-ren legearen bertsio hau soilik beteko da zehazki zirkuitu itxia sekzio arbuiagarriko begizta bat denean, eta beste kasu batzuetan ez da baliozkoa izango. Batetik ez du kontuan hartzen Faradayren legea fluxuaren deribatu totalari eta ez partzialari dagokiola eta, bestetik, IEEa ez dagoelako nahitaez mugatuta bide itxi batera, osagai erradialak ere izan dezakeelako, beherago azaltzen den moduan. Beste bertsio bat, Maxwell-Faraday ekuazioa (behean azaltzen duguna), baliozkoa da kasu guztietan, eta Lorentz indarreko legearekin batera erabiltzen bada bat dator Faraday-ren legearen aplikazio zuzenarekin.

Outline of proof of Faraday's law from Maxwell's equations and the Lorentz force law.

Consider the time-derivative of flux through a possibly moving loop, with area $\Sigma (t)$: ${\frac {d\Phi _{B}}{dt}}={\frac {d}{dt}}\int _{\Sigma (t)}\mathbf {B} (t)\cdot d\mathbf {A}$ The integral can change over time for two reasons: The integrand can change, or the integration region can change. These add linearly, therefore: $\left.{\frac {d\Phi _{B}}{dt}}\right|_{t=t_{0}}=\left(\int _{\Sigma (t_{0})}\left.{\frac {\partial \mathbf {B} }{\partial t}}\right|_{t=t_{0}}\cdot d\mathbf {A} \right)+\left({\frac {d}{dt}}\int _{\Sigma (t)}\mathbf {B} (t_{0})\cdot d\mathbf {A} \right)$ where t0 is any given fixed time. We will show that the first term on the right-hand side corresponds to transformer EMF, the second to motional EMF. The first term on the right-hand side can be rewritten using the integral form of the Maxwell–Faraday equation: $\int _{\Sigma (t_{0})}\left.{\frac {\partial \mathbf {B} }{\partial t}}\right|_{t=t_{0}}\cdot d\mathbf {A} =-\oint _{\partial \Sigma (t_{0})}\mathbf {E} (t_{0})\cdot d{\boldsymbol {\ell }}$ Area swept out by vector element dℓ of curve ∂\Sigma in time dt when moving with velocity v. Next, we analyze the second term on the right-hand side: ${\frac {d}{dt}}\int _{\Sigma (t)}\mathbf {B} (t_{0})\cdot d\mathbf {A}$ This is the most difficult part of the proof; more details and alternate approaches can be found in references. As the loop moves and/or deforms, it sweeps out a surface (see figure on right). The magnetic flux through this swept-out surface corresponds to the magnetic flux that is either entering or exiting the loop, and therefore this is the magnetic flux that contributes to the time-derivative. (This step implicitly uses Gauss's law for magnetism: Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.) As a small part of the loop $d{\boldsymbol {\ell }}$ moves with velocity v for a short time $dt$, it sweeps out a vector area vector $d\mathbf {A} =\mathbf {v} \,dt\times d{\boldsymbol {\ell }}$. Therefore, the change in magnetic flux through the loop here is $\mathbf {B} \cdot (\mathbf {v} \,dt\times d{\boldsymbol {\ell }})=-dt\,d{\boldsymbol {\ell }}\cdot (\mathbf {v} \times \mathbf {B} )$ Therefore: ${\frac {d}{dt}}\int _{\Sigma (t)}\mathbf {B} (t_{0})\cdot d\mathbf {A} =-\oint _{\partial \Sigma (t_{0})}(\mathbf {v} (t_{0})\times \mathbf {B} (t_{0}))\cdot d{\boldsymbol {\ell }}$ where v is the velocity of a point on the loop $\partial \Sigma$. Putting these together, $\left.{\frac {d\Phi _{B}}{dt}}\right|_{t=t_{0}}=\left(-\oint _{\partial \Sigma (t_{0})}\mathbf {E} (t_{0})\cdot d{\boldsymbol {\ell }}\right)+\left(-\oint _{\partial \Sigma (t_{0})}(\mathbf {v} (t_{0})\times \mathbf {B} (t_{0}))\cdot d{\boldsymbol {\ell }}\right)$ Meanwhile, EMF is defined as the energy available per unit charge that travels once around the wire loop. Therefore, by the Lorentz force law, $EMF=\oint \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\cdot {\text{d}}{\boldsymbol {\ell }}$ Combining these, ${\frac {d\Phi _{B}}{dt}}=-EMF$

erreferentziak

[[Kategoria:Elektrodinamika]]