A question about the right hand rule

[Srullic]

Hey everyone, I was hoping you could help me solve a little problem. I'm having a little difficulty understanding why the right hand rule works for determining the direction of the magnetic field. I've found a similar question posted here:

https://www.physicsforums.com/threads/why-does-the-right-hand-rule-works.472162/

And I feel that while the answers were insightful, they kind of missed the point of the question, so I'll try to state mine in a way that can be answered in a yes -or-no manner.

Suppose I have an infinately long conductive wire, and I want to determine the magnetic field around it using only maxwell's equations. Suppose I already deduced the strength of the field, and the fact that it is perpendicular to the direction of both the current and the distance from the wire (I know it is possible to do because I have done it).
My question is this: can I deduce, only from maxwell's equations, whether the field will be directed into the plane of the paper or out from it? Or do I have to experimentally place a test charge close to the wire and measure the Lorentz force on it?

In short, is the right hand rule a property of the magnetic field stemming directly from the mathematical description of it, or is it a purely empirical property?

 

[Doc Al]

My question is this: can I deduce, only from maxwell's equations, whether the field will be directed into the plane of the paper or out from it?

Yes.

 

[Srullic]

Yes.

Elaboration please...?

 

[Dale]

I'll try to state mine in a way that can be answered in a yes -or-no manner.

Yes.

Elaboration please...?

Perhaps you could describe in a less "yes-or-no manner" what type of elaboration you are interested in that is not covered in the previous thread and not addressed by the "yes" that it seems that you specifically requested.

 

[Srullic]

Perhaps you could describe in a less "yes-or-no manner" what type of elaboration you are interested in that is not covered in the previous thread and not addressed by the "yes" that it seems that you specifically requested.

Well I didn't think I would be taken so literally. If I'm writing a long-ass question, I'll expect people to reply me with a thoughtful answer.
What I meant was, if the answer is yes then show me the proof, and if the answer is no then explain why this is not possible.
When I said the question is asked in a yes-or-no manner, I meant that the question has a definitive answer, as opposed to a vague question like 'why does the right hand rule work?'.

 

[Doc Al]

Well, I did give a definitive 'yes or no' answer, as requested.

A more elaborate answer depends on how much you understand about Maxwell's equations, specifically the one referred to as Ampere's Law. All the sign conventions, such as the right hand rule in question, are embedded in the mathematics. I'll just give you a taste.

Applied to your example, that equation can be written as:
$$\nabla \times \vec{B} = \mu_{0}\vec{J}$$
or
$$\oint \vec{B} \cdot \vec{dr} = \mu_0 \ I_{enclosed}$$
The first version relates the "curl" of the magnetic field to the current. Note that the curl has embedded in it a cross product, which of course implies a right hand rule.

The second version, the integral form, relates a line integral around a closed loop (the "circulation") to the enclosed current. By convention, the orientation and sign of such an integral follows a right hand rule.

 

[Srullic]

Heah, I'll admit it was poor wording on my part.
Anyway I think I got it, thanks a lot!