# Why doesn't Faraday's Law include speed of EM wave

$$\nabla\times \overrightarrow{E} = -\frac{d\overrightarrow{B}}{dt}$$ Faraday's Law says: any change in the magnetic field causes circulation of electric field.

$$\mathbf{\nabla \times B} = \mu_0 \mathbf{j} + \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$$ Maxwell's Law says: any change in the electric field causes circulation of magnetic field.

Right?

1-) If it is right, then why doesn't Faraday law have a speed of light parameter but Maxwell has?

2-) Is it because Faraday is all about the electric current in the wire and Maxwell's Law is about vacuum?

3-) Is that the difference between Maxwell's and Faraday's Law?

• Here is the best explanation of Faraday/Maxwell laws ever: www.irregularwebcomic.net/1420.html – Jim Green Oct 23 '15 at 19:37

## 1 Answer

If you work in Gaussian units, where the electric and magnetic field appear on the same footing and have the same units, Faraday's law does contain a factor of $1/c$

$\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}$.

In Gaussian units, Ampere's equation takes the form

$\nabla \times \mathbf{B} = \frac{4\pi}{c}\mathbf{J} + \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t}$,

where there is now single factor of $c$ in the denominator of both terms on the right-hand side. The advantage of Gaussian units is that they emphasize the fact that $\mathbf{E}$ and $\mathbf{B}$ come together to form the single electromagnetic field (sometimes written $F_{\mu\nu}$).

The real difference between Faraday's and Ampere's equations is the lack of a "magnetic monopole current" in Faraday's law. Despite the symmetry that such a term would add to the equations, no compelling experimental evidence for monopoles has ever been found.

• I'm surprised by your answer. Is that formula founded by Faraday himself? I mean was he the first scientist that expressed speed of light? Or was this formula modified later (by Maxwell or some other scientists)? Thanks – user50322 Oct 23 '15 at 17:10
• I'm not an expert on the history of Maxwell's equations, so I cannot say who first wrote down the equations in this form. Note that the equations in my answer are completely equivalent to those in your question; they are just expressed in different system of units. Units can appear somewhat subtle in electromagnetism. If you're unfamiliar with Gaussian units, I'd recommend looking at the Wikipedia page The physics is completely the same. – wijay Oct 23 '15 at 17:56
• @user50322: Ahem, Wikipedia: “Speed of light”/History – Holger Oct 23 '15 at 18:00
• @user50322 The fact is, you can't get the speed of light without both equations being combined into the wave equation. When you write them separately, the speed of light can be absorbed into the definition of units for electric and/or magnetic field, as evidenced by how it appears in different places in different unit systems. We only know how to extract the speed of light ex post facto once we've used it to define our units self-consistently. – user10851 Oct 23 '15 at 18:10