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I have been trying to find out who discovered Maxwell-Faraday equation, $\nabla\times E=-\frac{\partial B}{\partial t}$. Was it Maxwell himself, or was it Franz Ernst Neumann who derived it?

The equation means that a time varying magnetic field induces a spatially varying electric field.

While searching for the answer I stumbled upon an important piece which is given below.

"Maxwell's unique contribution was his "Ampère's" circuital law (both forms: with and without the displacement current term). Faraday's Law was derived by Franz Neumann in $1845$. Gauss (and Lagrange before him) knew the compatibility between Coulomb's law and "Gauss"'s law. I'm not sure who was the first to derive $\nabla\cdot B=0$".– Geremia Source: Why is Maxwell and not Ampère credited for unifying electricity and magnetism?

Faraday's law in itself was more about voltage generated as a result of magnetic field. To confuse things more, it is said that it was Neumann who first put the result of Faraday's experiment into a mathematical form somewhere around $1845$. Why did it take so long to mathematize the result of an experiment? Why didn't Faraday himself converted the result of his experiment(s) into a mathematical form?

Related discussion: Who discovered the magnetic vector potential, $\vec{A}$? "The laws of induction of electric currents in mathematical form was established by Franz Ernst Neumann in 1845." -https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction

Could you please help me to find the answers to these queries? Thanks for your help!

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    $\begingroup$ The wiki article you have linked to states that it was Heaviside who first mathematized Faraday's ideas into the form in which they appear in Maxwell's equations. As to why Faraday himself did not do the maths, I believe he simply lacked the mathematical skills to do so, having little-to-no formal education. Mind you, Heaviside himself was largely self-taught. $\endgroup$
    – nwr
    Jul 14, 2019 at 17:47
  • $\begingroup$ @NickR: Thank you. I believe that you were referring to this part from the Wikipedia article, "In Maxwell's papers, the time-varying aspect of electromagnetic induction is expressed as a differential equation which Oliver Heaviside referred to as Faraday's law even though it is different from the original version of Faraday's law, and does not describe motional EMF. Heaviside's version (see Maxwell–Faraday equation below) is the form recognized today in the group of equations known as Maxwell's equations.". I have read that it was Heaviside who converted Maxwell's, I think, 22, cont'd $\endgroup$
    – PG1995
    Jul 15, 2019 at 5:18
  • $\begingroup$ EM equations into a set of only 4 using his vector notation. It looks like the contribution of Heaviside came later after Maxwell had already formulated his equations. According to the same article, it was Neumann who established laws of induction of electric currents into mathematical form in 1845. The question then becomes if it was Neumann or Maxwell who formulated the so called Maxwell-Faraday equation such that it stated that a time varying magnetic field induces a spatially varying electric field. Perhaps, Neumann's equation was only able to describe EMF generation in a wire etc. Thanks. $\endgroup$
    – PG1995
    Jul 15, 2019 at 5:29
  • $\begingroup$ Electromagnetism and mathematics were growing and feeding each other at the same time. The concepts of grad, divergence, curl, (and Green's functions, potentials, etc.) were being formed at this time (by Weber, Gauss, Neumann) as localized versions of the integral laws. See en.wikipedia.org/wiki/Weber_electrodynamics $\endgroup$ Jul 17, 2019 at 7:43
  • $\begingroup$ Chrystomath, thank you but it doesn't really answer the main question who discovered the so-called Maxwell-Faraday equation. Interestingly, on this page, hsm.stackexchange.com/questions/2935/… , Omar Nagib says "I'm aware that Maxwell generalized Faraday's law and Ampère's law by adding displacement current, ...?" The answer by @Geremia is interesting but only addresses Ampere's law. It looks like it was Maxwell himself who formulated the given equation which stated that a time varying magnetic field induces a spatially varying electric field. $\endgroup$
    – PG1995
    Jul 18, 2019 at 2:01

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To my knowledge, Neumann gave the formula in 1845.It is said that Faraday did not provide any mathematical formulas. The law of electromagnetic induction was obtained by Neumann in 1845 by calculating the energy between two coils based on the Ampere force formula. $$Energy=-\frac{I_{1}I_{2}}{2}\oint_{C_{2}}\oint_{C_{1}}\frac{d\boldsymbol{l}_{1}\cdot d\boldsymbol{l}_{2}}{r}\ \ \ \ (1)$$ From this we can obtained that $$\mathcal{E}_{2,1}=-\frac{\mu_{0}}{4\pi}\oint_{C_{2}}\oint_{C_{1}}\frac{dI_{1}}{dt}\frac{d\boldsymbol{l}_{1}\cdot d\boldsymbol{l}_{2}}{r}\ \ \ \ (2)$$ I do not clear how can we from (1) to Derived (2). But when we obtained (2) it is easy to obtained the Faraday law. First define Induced electromotive force and magnetic vector potential $$\mathcal{E}_{2,1}\triangleq\oint_{C_{2}}\boldsymbol{E}_{1}\cdot d\boldsymbol{l}_{2}$$ $$\boldsymbol{A}_{1}\triangleq\frac{\mu_{0}}{4\pi}\oint_{C_{1}}\frac{I_{1}d\boldsymbol{l}_{1}}{r}$$ Hence we obtained, $$\oint_{C_{2}}\boldsymbol{E}_{1}\cdot d\boldsymbol{l}_{2}=-\oint_{C_{2}}\frac{\partial}{\partial t}\boldsymbol{A}_{1}d\boldsymbol{l}$$ or $$\oint_{C_{2}}(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})\cdot d\boldsymbol{l}_{2}=0$$ or $$\iint_{\Gamma}\nabla\times(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})\cdot\hat{n}d\Gamma=0$$ or $$\nabla\times(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})=0$$ or $$\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1}=-\nabla\phi_{1}$$ or $$\boldsymbol{E}_{1}=-\nabla\phi_{1}-\frac{\partial}{\partial t}\boldsymbol{A}_{1}$$ or omit the subscript, $$\boldsymbol{E}=-\nabla\phi-\frac{\partial}{\partial t}\boldsymbol{A}$$ This is Faraday's law proposed by Maxwell himself. Maxwell's descendants further rewrote this formula as, $$\nabla\times\boldsymbol{E}=-\nabla\times\nabla\phi-\frac{\partial}{\partial t}\nabla\times\boldsymbol{A}$$ or $$\nabla\times\boldsymbol{E}=-\frac{\partial}{\partial t}\nabla\times\boldsymbol{A}$$ define $$\boldsymbol{B}\triangleq\nabla\times\boldsymbol{A}$$ we obtained $$\nabla\times\boldsymbol{E}=-\frac{\partial}{\partial t}\boldsymbol{B}$$ We see that the Faraday's law of electromagnetic induction derived from Neumann's electromagnetic induction formula is mediocre, which we are using in today's textbook. Therefore, this credit belongs to Neumann The vector potential we also defined above is also known as Neumann vector potential. Considering, $$\oint_{C}\cdots Idl\rightarrow\int_{V}\cdots JdV$$ Obtain, $$\boldsymbol{A}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{\boldsymbol{J}}{r}dV$$ It is worth mentioning that Weber also proposed the law of electromagnetic induction in 1846, and the vector corresponding to this law of electromagnetic induction is, $$\boldsymbol{A}_{W}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{(\boldsymbol{J}\cdot\boldsymbol{r})\boldsymbol{r}}{r^{3}}dV$$ If the coil is a closed loop, these two magnetic vector potentials are equivalent, that is, the induced electromotive force calculated by them is equal. Therefore, Weber is also another contributor to the law of electromagnetic induction.

It is worth mentioning that Maxwell defined a magnetic field based on the curl of a magnetic vector under quasi-static conditions $$\boldsymbol{A}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{\boldsymbol{J}}{r}dV\ \ \ \ \ (3)$$ Because, $$\nabla\times\boldsymbol{A}=\frac{\mu_{0}}{4\pi}\int_{V}\nabla(\frac{1}{r})\times\boldsymbol{J}dV$$ or $$\nabla\times\boldsymbol{A}=\frac{\mu_{0}}{4\pi}\int_{V}\boldsymbol{J}\times\frac{\boldsymbol{r}}{r^{3}}dV\ \ \ \ (4)$$ From Biota's law $$\boldsymbol{B}=\frac{\mu_{0}}{4\pi}\int_{V}\boldsymbol{J}\times\frac{\boldsymbol{r}}{r^{3}}dV\ \ \ \ (5)$$ Compare (4) and (5) to obtain the definition, $$\boldsymbol{B}\triangleq\nabla\times\boldsymbol{A}$$

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  • $\begingroup$ I would like to add that there are many predecessors to this Faraday's law like Felici and Lenz. $\endgroup$
    – Mauricio
    Jan 9 at 16:53
  • $\begingroup$ I agree it. Felici and Lenz did the contribution to Faraday's law, but the work of mathematic part belong to Neumann and Weber. $\endgroup$
    – ShRenZhao
    Jan 9 at 19:45
  • $\begingroup$ As for $\mathbf B =\nabla \times \mathbf A$ wasn't it Lord Kelvin's? $\endgroup$
    – Mauricio
    Jan 10 at 8:17
  • $\begingroup$ @Mauricio It is correct, $\boldsymbol{B}=\nabla\times\boldsymbol{A}$ was found by Lord Kelvin (William Thomson). But people gave the contribution of magnet vector potential to Neumann. I think that means these two formula $\boldsymbol{B}=\nabla\times\boldsymbol{A}$ and $\boldsymbol{E}=-\frac{\partial}{\partial t}\boldsymbol{A}$ also belong to the contribution of Neumann. I am not very sure. $\endgroup$
    – ShRenZhao
    Jan 10 at 16:58

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