Do magnetic fields interact with each other to attract or repel each other?

Question

When determining if magnets will attract or repel each other, we observe the shape of their magnetic fields and hence poles (The image below shows this set up). In the below image, I explain it as the magnetic field arrows flowing from the north pole of one magnet to the south pole of another which is why they attract each other.

Why can't I seem to apply the same analysis to determine if two straight wires that are carrying current (the experiment Ampere performed) will attract or oppose each other. Why does it seem like I can only use Lorentz's law where a field is interacting on moving charged particles to understand what will happen?

Image Sources: https://byjus.com/physics/magnetic-field/ https://www.quora.com/What-happens-when-two-parallel-wires-carry-an-electric-current-in-the-same-direction

Answer 1

The lines are visualisations to represent a vector field.
At each point in space there is a magnetic field strength and a direction for that field.

Historically the magnetic flux density was the number of field lines per unit area and that is were the term flux (= flow) comes from with magnetic flux being the total number of lines.
You will still find lots of textbooks which are in esu, emu, cgs and Gaussian units from a time when there were also magnetic poles which followed an inverse square law just like Coulomb's law for electric charges.

The magnetic field lines are visualisations and so you have some degree of artistic license with them provided you follow the simple properties:

• Start and finish on themselves although it is often much clearer if you have them starting on a North pole and finishing on a South pole.

• The arrow on a magnetic field line goes away from a North pole and goes towards a South pole or follows the right hand grip rule for currents.

• Magnetic field lines are in a state of tension. That is why a North pole attracts a South pole!

• Magnetic field lines never cross and repel each other. That is why two North poles repel one another!

• The closer the lines are to one another the stronger is the magnetic field (magnetic flux density).

So now look at the field patterns between two current carrying parallel conductors.

From diagram a you can infer that it attraction because the field lines which go around both conductors are in a state if tension and there are few field lines between the two conductors for them to produce a large enough repulsive force.

From diagram b you can infer that there is repulsion because the field lines between the two conductors are repelling each other.

From your magnets diagram one might infer there is attraction because the field lines between the poles are in a state of tension.

Answer 2

Actually, there are 2 types of magnetic interactions which seem to be very different on the phenomenlogical level (but have the same origin on the microscopical level):

1) magnetic interaction between moving charges

A electric current --- thus moving charges -- (or a (fast) changing electric field) is needed for the generation of the magnetic field according to Ampere's circuital law. Second the created magnetic field only acts on moving charges (not on charges at rest) according to Lorentz' law ($\mathbf{v}$ is the velocity of the moving charge):

$$\mathbf{F} = e(\mathbf{v}\times \mathbf{B})$$

In this respect magnetic interaction is different from electrical interaction where a charge (an electric monopole) at rest can already generate an electrical field (whereas magnetic monopoles do not exist respectively have not been seen yet) and can act on charges at rest.

This is the origin of the magnetic forces between 2 wires.

2) mutual interaction of magnetic dipoles

(A permanent magnet is a magnetic dipole)

Sources distributed in space of the magnetic or electric field can be developed in a series of monopoles, dipoles, quadrupoles etc. It is actually a Taylor series which
helps to classify sources of the magnetic field of different distribution. Particularity of the magnetic case is that monopoles do not exist. In different words, the first term in the mentioned series is zero. The force between magnetic dipoles is based on

$$\mathbf{F} = \nabla(\mathbf{m}\cdot \mathbf{B}) \tag{1} $$

where $\mathbf{m}$ is the magnetic moment and $\mathbf{B}$ is the magnetic induction (also called magnetic flux density). Actually the $\mathbf{B}$ can be or any origin, but if it comes from another magnetic dipole it can be computed by:

$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\left[ \frac{3\mathbf{r}(\mathbf{m}\cdot \mathbf{r})}{r^5} - \frac{\mathbf{m}}{r^3}\right] \tag{2}$$

where $\mu_0$ is the vacuum permeability and $\mathbf{r}$ is the vector between the center of the dipole and the point where magnetic induction is required. (2) inserted in (1) yields a quite complicated expression that can be simplified under certain circumstances. For instance if the poles of the dipole is small enough, then each pole can be considered as a magnetic point charge which makes the full expression for the magnetic force $\mathbf{F}$ much simpler.

The same interaction also exists for electrical dipoles although it is less frequent in daily life, but quite common between molecules for instance.

This is the origin of the magnetic force between 2 magnetic dipoles respectively permanent magnets.

Actually, there would be a lot of stuff to be said additionally to this post, above all the microscopic origin of the magnetization of material, in particular a permanent magnet, how to take correctly account for the effects of magnetization of material etc. But this would be out of scope of the given question. Anyway, wikipedia is an excellent source to get an overview on these topics.