Understanding of Gauss law using vector fields

I was going through the conventions and terminologies followed to describe the magnetic interactions. I understood that the field lines are just a simpler representation of the magnetic interaction described in terms of a vector field.

So those field lines basically relates to the line of force due to magnetic interactions. I hope this understanding is right.

And the magnetic flux is defined as net magnetic field lines crossing the area. And so flux indirectly relates to the magnetic forces.

After all these understandings I considered Gauss law of magnetism which states that the total magnetic flux through a closed surface is equal to zero. The observation that magnetic monopoles do not exist supports this law.

Without involving surface integrals concept (Area vector) why can't I say If the flux is zero, Field lines are zero? And if yes Can I say that the magnetic force inside a closed surface is zero?

But this conflicts with my general understanding of the field line. Why should a close surface affect the force due to magnetic interaction?

If the surfacae integral is necessary to answer to this Question why the surface area vector is related to force vector in Gauss Law?

• Magnetic force is not along the magnetic field lines. $\mathbf F=q\mathbf v\times\mathbf B$. The force is actually perpendicular to the field lines. – Aaron Stevens Jul 11 at 18:21
• @probably_someone comment battle – Aaron Stevens Jul 11 at 18:22
• Also, Gauss's Law of Magnetism is a consequence of the fact that we have never observed magnetic monopoles to exist. It doesn't support that fact; support for that fact comes from observations. – probably_someone Jul 11 at 18:23
• Magnetic force doesn't have a vector field equivalent, at least not in $\mathbb{R}^3$, because magnetic force on a charged particle is also perpendicular to that particle's velocity (and, in fact, a stationary charged particle will experience no magnetic force regardless of the magnetic field at its location). In $\mathbb{R}^6$ you might be able to construct such a vector field (three dimensions for position, three dimensions for velocity), but that may not be what you're looking for. – probably_someone Jul 11 at 18:30
• Because the field lines show you what the magnetic field looks like. The point we're trying to get across is that the relationship between magnetic field and magnetic force is more complicated than you assume. – probably_someone Jul 11 at 18:34

why can't I say If the flux is zero, Field lines are zero?

The flux through a particular area is the difference of the number of magnetic field lines pointing one way through an area and the number of magnetic fields pointing the other way. If you have the same number of field lines pointing into and out of a particular surface, then the flux will be zero even if the field is nonzero. For an analogy, it isn't correct to say: if $$x-y=0$$, then $$x$$ and $$y$$ must both be $$0$$. There are many combinations of $$x$$ and $$y$$ that will satisfy that equation, just like there are many nonzero field configurations that will give you zero flux through a particular surface.

Why should a close surface affect the force due to magnetic interaction?

Applying Gauss's Law for Magnetism closed surface is a mathematical tool that we use to describe how a magnetic field looks. It often simplifies what would otherwise be a difficult calculation; however, it is by no means the only way that we have to calculate the magnetic field (for example, the Biot-Savart Law is another useful tool). It doesn't affect anything about the field (and certainly doesn't affect the force, since the relationship between magnetic field and magnetic force is complicated).

• This clear some of doubts. But in first place why should we describe how a field is going to look if it's relation to magnetic force is complicated? – VKJ Jul 11 at 18:49
• @VKJ Because the magnetic field is necessary to describe the magnetic force. It just isn't a complete description of magnetic force - for example, for the force on a charged particle, we also need its velocity. – probably_someone Jul 11 at 18:51
• So can I say that for a given velocity the points in area where flux is more will experience more force? – VKJ Jul 11 at 18:54
• @VKJ It isn't really correct to talk about the flux at a particular point. Flux is defined as being through an area of a particular size. If you want to talk about the magnetic field at a particular point, then you should look at the flux density at that point (in other words, how much the flux through a tiny area about that point changes if that area is increased by a tiny amount). In fact, magnetic field is sometimes referred to in older books as "magnetic flux density" for that reason. – probably_someone Jul 11 at 18:58
• And I am still not understanding the importance of taking surface Integral of a visualisation tool - the field lines. And how the direction of field lines are know? – VKJ Jul 11 at 18:59