Question about Maxwell's Equations

[lugita15]

Is it possible to derive Faraday's Law from the other three Maxwell equations plus the conservation of charge? If so, how?

Any help would be greatly appreciated.
Thank You in Advance.

 

[Xezlec]

Conservation of charge is not related to Faraday's law in any way, since Faraday's Law has absolutely nothing to do with charge. Faraday's law only talks about fields.

Faraday's Law cannot be derived from the other equations. If it could, it wouldn't be considered one of Maxwell's equations.

 

[Meir Achuz]

If you take the divergence of Maxwell's curl H equation, you get the continuity equation, which is equivalent to conservation of charge (if you use the div D equation). But you can't go the other way, although that is probably how Max deduced the D dot term.

 

[Meir Achuz]

You can go the other way, to some extent.
1. Start with the curl H equation with only the j term on the right.
2. Take the divergence of both sides.
3. This gives div j=0, but div j =-d rho/dt.
4. This requires adding the d D/dt term (using the div D Maxwell eq.) , which is equivalent to Farady's law.

 

[Xezlec]

You can go the other way, to some extent.
1. Start with the curl H equation with only the j term on the right.
2. Take the divergence of both sides.
3. This gives div j=0, but div j =-d rho/dt.
4. This requires adding the d D/dt term (using the div D Maxwell eq.) , which is equivalent to Farady's law.

How is that equivalent to Faraday's law? Faraday's law relates d B/dt to curl E, not d E/dt to curl B. Doesn't it?

 

[arunma]

My understanding is this: despite their mathematically rigorous statements, Maxwell's Equations are all empirical laws, meaning that they can't be derived by any axiomatic approach. As such, the only way to "derive" Faraday's Law would be to do a physical experiment and deduce it. In this case, you'd need to alter the magnetic flux of a conducting loop, and show that the line integral of the electric field (=the EMF) is equal to the rate of change of flux through the loop. But since each of Maxwell's equations say different things about the electric and magnetic fields, I can't think of a way in which you could derive any of them from the others.

 

[Meir Achuz]

How is that equivalent to Faraday's law? Faraday's law relates d B/dt to curl E, not d E/dt to curl B. Doesn't it?

Yes, I just got careless.