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My question is, why is the magnetic force going the same way as the fingers, why not the other way. What causes the magnetic field to choose a direction if everything is symetrical

semir213
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  • It has to do with the relation between the electrons magnetic dipole moment and the intrinsic spin. A positron has both properties too, but anti-parallel to this properties of an electron. That is the reason why these particles get deflected in opposite directions under the influence of an external magnetic field. To understand this read about gyroscopic effect. – HolgerFiedler Apr 15 '16 at 03:41

2 Answers2

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Might be an elementary answer to your question. But if you look at the Biot and Savart Law, you will see that the magnetic field generated depends on the cross product between the position vector and the "current loop" vector. This vector of course points in a direction that is perpendicular to both vectors. The symmetry is not broken, there are already 2 direcctions specified in this context.

Now, if your question is why right hand rule and not left hand rule, then this is simply convention, just like we decide to call one type of charge positive and the other negative and not viceversa, e.g we might call all negative magnetic field positive and use the left hand rule.

Stefano
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The convention chosen for the cross product defines the direction of the magnetic field.

You are interested in the observable effects of the magnetic field, like the force on a particle in motion. The force should be given by $$\mathbf F = q\mathbf v\times\mathbf B \tag{1}$$ where $q$ is the charge, $\mathbf v$ is the velocity, and $\mathbf B$ is the magnetic field. There is no ambiguity in $\mathbf v$ or $\mathbf F$, but this is the definition (operationalization) of $\mathbf B$. For that purpose, it is ambiguous to the extent that the cross product is. However, since $\mathbf v\times_R \mathbf B = \mathbf v \times_L (- \mathbf B)$ the convention you choose for the cross product only affects the direction of the magnetic field.

In a deep sense what is going on mathematically is that the magnetic field isn't really a vector and the cross-product is an inelegant barbaric relic. That the cross product is only defined in 3 space dimensions should be a sign that (1) isn't the best way to talk about the magnetic field and force. In fact, in dimensions other than 3 the magnetic field isn't even a vector. It's something else called a two-form or bivector. It just so happens that in 3 dimensions, vectors and two-forms (bivectors) are equivalent - up to an ambiguity in sign that corresponds to the left-handed or right-handed convention. When expressed in terms of forms, (1) is manifestly independent of any such conventions.

Robin Ekman
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