RHR for Magnetic fields, why is it design in such a way?

[Wek]

I don't feel like this is a homework question, more like a curiosity question. If I'm wrong then feel free to move it to the homework section and my apologies.

I know how the RHR works for magnetic forces and I can apply it with no problem, however I'm not a physics major so maybe I'm missing crucial info. Anyway, my question is why the RHR rule applied the way it is. I mean take a look at this picture about how the railgun works:

http://upload.wikimedia.org/wikipedia/commons/9/9a/Railgun-1.svg

For the projectile line (orange), why do I have to use the RHR (hand upright) in such a way that the force is facing away (towards the left)? I'm not questioning the mechanics of the device, it wouldn't make sense to have the force pointed at you for obvious reasons. But, why would applying the RHR with the hand pointing upside-down (therefore the force directed to the right instead) be wrong?

Thanks

 

[Matterwave]

The right hand rule is a convention so that we all define our magnetic fields the same way. We could have defined the convention using a left hand rule, as long as in the end the direction of the force stays unchanged. The force is the only thing we can "see", we can't "see" magnetic fields, and it's not like magnetic fields are actually little arrows that point in a certain direction. What we can see is how particles act in magnetic fields. As long as we don't change the equation of motion, it is arbitrary which direction I want to define magnetic fields. If the magnetic fields change direction (so that a current's magnetic field direction is defined by a left hand rule), I merely have to redefine how they act on plus and minus charges (also using the left hand rule).

 

[chogg]

I would merely like to add to Matterwave's excellent answer that it's possible (and desirable!) to avoid this arbitrary convention altogether. Instead of visualizing the magnetic field as a vector, pointing in some direction, it can be visualized as a plane with a given orientation (clockwise or counterclockwise). See this puzzle:
http://www.av8n.com/physics/pierre-puzzle.htm
and its solution:
http://www.av8n.com/physics/pierre-answer.htm
for a concise introduction.