# Deduce plus and minus with Cross Product in 3th and 4th Maxwell equations

The laws:

$\nabla \times \bar{E} = \bar{I}_{m} - \frac{\partial \bar{B}}{\partial \bar{t}}$

$\nabla \times \bar{H} = \bar{J}_{f} + \frac{\partial \bar{D}}{\partial \bar{t}}$

so how can I remember with fingers or any other deduction method which is minus and which is plus? Since there is now the nabla, I am a bit lost how the cross product hand-rule work.

The term $\bar{I}_{m} = 0$ if magnetic monopoles do not exist. It is there to show that the formulas are of the same structure, cannot just remember which is minus and which is plus.

[Update]

Is it easier to deduce the laws if I suppose that the circuit moves and not the field?

$\bar{v} \times \bar{E} = \bar{I}_{m} - \frac{\partial \bar{B}}{\partial \bar{t}}$

$\bar{v} \times \bar{H} = \bar{J}_{f} + \frac{\partial \bar{D}}{\partial \bar{t}}$

I am unsure whether the formulae are right so please check it. Could this way result in some easy deduction?

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Wouldn't this be more appropriate for SE.physics? – Fabian May 18 '11 at 15:13
Regarding the right hand rule: In Fourier space $\nabla \Rightarrow i \vec{k}$, so then your right hand rule works again... Regarding the sign: Which one is plus and which is minus is in fact convention. It is just important that they are different. – Fabian May 18 '11 at 15:14
@Fabian: I was about to post it to physics but they do not have tag "cross-product", I am more interested in this topic mathematically than just conventionally. I am looking some proper reusable way to memorize/deduce it without guessing. Suppose I choose the sign wrong, what kind of mathematical corrections have to I look for to correct the mischief? If I have the sign wrong way, is there some way mathematically to see it with some analysis that it is wrong? I can understand the oddity but physical memorizing is way above my head. – hhh May 18 '11 at 15:28
@hhh: there is nothing wrong with the mathematics when you get a sign wrong. There might be however something wrong with the physics. I still believe it does not belong here and you should post it in physics. – Fabian May 18 '11 at 15:32
@Fabian: could you elaborate on it? It may help me to get the signs right. – hhh May 18 '11 at 15:32

## 1 Answer

Dear hhh, first of all, magnetic "monotones" are called "monopoles".

Second, you can't replace $\nabla \times$ by $\vec v\times$ in Maxwell's equations because these two objects don't even have the same units.

Third, you must just remember the signs. The Ampere's law, $\nabla\times H= j$, has the plus sign. It's the oldest law mixing electricity and magnetism, so it has the plus sign. It's about the magnetic fields around electric wires. The corresponding currents and magnetic fields around them are described by the right-hand rule.

You may also remember that in this oldest law - magnetic fields of wires - Maxwell's correction $\partial D/\partial t$ also comes with a plus sign.

It then follows that Faraday's law must have the opposite sign - so there is $-\partial B / \partial t$ on the right hand side for the equation for $\nabla \times E$.

But more generally, it's meaningless for you to incorporate "right hand rules" into the form of Maxwell's equations. Right hand rules are not supposed to help you to remember signs in a written form of the equations - on the paper; the "plus rules" and "minus rules" will do.

Right hand rules are supposed to help you to find the directions of vectors in various situations in the real world where there is no God-given understanding which direction in a given setup is "plus" and which direction is "minus". As some people have mentioned before me, "hands" are only useful to define "rules" when you deal with a physical world, so hands are not useful to write down equations in mathematics.

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is it not possible to derive things such as Lorentz force $\bar{F} = q(\bar{E}+ \bar{v} \times \bar{B})$ with Maxwell equations and use such derived results as a backup mnemonic to remember the oddity? Feuman mentioned, Feuman Lectures about Physics -book or s/thing like that, that the term in Lorentz force and the fourth Maxwell equation had some interesting relationship, trying to dig that part... – hhh May 18 '11 at 18:39
@hhh: the physicist is Richard Feynman – Ross Millikan May 18 '11 at 19:55