How do magnetic fields behave in superposition?

[aspodkfpo]

Homework Statement:: n/a
Relevant Equations:: n/a

These are the answers diagrams, with my questions in red.
1. In arrangement 1, I was wondering why there isn't any magnetic lines inside the magnet. While it seems that vector addition would make the field go outside the magnetic-less rectangle, isn't the right hand sides magnetic field at that point stronger, or am I wrong/do we ignore this?
(in blue), when imagining the superposition of fields do the field lines get drawn like they go through the magnets (2. with the inside parts excluded or included in the final diagram?) or 3. does it look different from the other side which does not have magnetic field blocking its path?
4. In arrangement 2, I was wondering whether that line breaks off into two at that point, or how does it work?
5. When doing superposition diagrams, are we expected to draw more loops than shown in the answers here?

 

[Merlin3189]

In arrangement 2: the liine does not split into two.
In arrangement 1: if the red line is drawn, it must touch the magnets at different points from any other line.
Because: at any point there can be only one line.
The lines are mathematical abstractions - like contour lines. They show at any point what the direction of the field is. So it can't have two directions at any point.

There may be points where the field does not know which way to go - eg. on the centreline, midway between two equal N poles. We sometimes put an x there, but no line.

Put enough lines to show the important features, or the features you want to show.
We show stronger or weaker fields by the density of lines in a region, so if you wat to show this, you may need several lines that go through the region of strong field.
But it's very difficult to sketch lines accurately (IMO), so the more you put, the more problems you have (IMO).

We often draw field lines through pieces of iron, or inside coils and solenoids, so I don't see why we can't draw them through magnets. I'd never really noticed before that we don't. I suppose it would cause confusion, because the lines would appear to go in the wrong direction. (Unless you also drew lots of NS pairs inside the magnet. Then, would you draw the lines through them!)
I suppose you have to realize that these lines were invented by Faraday as an aid to understand what happens. So draw the bits that aid understanding and don't draw ones that confuse.

 

[aspodkfpo]

In arrangement 2: the liine does not split into two.
In arrangement 1: if the red line is drawn, it must touch the magnets at different points from any other line.
Because: at any point there can be only one line.
The lines are mathematical abstractions - like contour lines. They show at any point what the direction of the field is. So it can't have two directions at any point.

There may be points where the field does not know which way to go - eg. on the centreline, midway between two equal N poles. We sometimes put an x there, but no line.

Put enough lines to show the important features, or the features you want to show.
We show stronger or weaker fields by the density of lines in a region, so if you wat to show this, you may need several lines that go through the region of strong field.
But it's very difficult to sketch lines accurately (IMO), so the more you put, the more problems you have (IMO).

We often draw field lines through pieces of iron, or inside coils and solenoids, so I don't see why we can't draw them through magnets. I'd never really noticed before that we don't. I suppose it would cause confusion, because the lines would appear to go in the wrong direction. (Unless you also drew lots of NS pairs inside the magnet. Then, would you draw the lines through them!)
I suppose you have to realize that these lines were invented by Faraday as an aid to understand what happens. So draw the bits that aid understanding and don't draw ones that confuse.
View attachment 267919

Wait so do the red lines exist on the 1st arrangement and does it not matter if they aren't drawn?

2. "Unless you also drew lots of NS pairs inside the magnet. Then, would you draw the lines through them!" -- what? If I drew lots of NS pairs how would the field even be able to travel through, does it not cancel each other out? Am I missing something? N S N S N S.

 

[Dale]

In arrangement 2, I was wondering whether that line breaks off into two at that point, or how does it work?

No, definitely not. Lines can never break off in two, combine into one, or intersect each other. The way it works is that, in a properly drawn field line diagram, there should be exactly the same number of lines that go off the diagram as that come back on to the diagram. Those lines pair up to form loops off the page. Since the lines do not cross, that means that you can find the "innermost" pairs (pairs 1 and 5 in this case) and know that those form a loop and then number the others from there. So in this case the line 4 coming in connects off the page with line 4 going out.

 

[etotheipi]

There is a lot of good stuff in @Dale's answer, you should read it carefully!

It is mentioned that field lines never split in two. This is because the magnetic field is uniquely defined at each point in space. Imagine a test charge $q$ with velocity $\vec{v}$ in the vicinity of a "hypothetical split" in a field line; what would the direction of the magnetic force $q\vec{v} \times \vec{B}$ be, out of the two choices? The resolution is of course that there can be no such "hypothetical split".

The second thing he mentioned was that the number of field lines coming into the diagram must be the same as the number of field lines going out. This comes from Maxwell II, i.e. $\nabla \cdot \vec{B} = 0$, which basically implies that through any closed surface (e.g. one enclosing your diagram) the net magnetic flux is zero.

 

[Merlin3189]

Wait so do the red lines exist on the 1st arrangement and does it not matter if they aren't drawn?

No they don't exist - anymore than any of the lines do. You draw the lines that help you.

2. "Unless you also drew lots of NS pairs inside the magnet. Then, would you draw the lines through them!" -- what? If I drew lots of NS pairs how would the field even be able to travel through, does it not cancel each other out? Am I missing something? N S N S N S.

Sorry to confuse you. This was just me trying to think of a possible way to draw the lines through the magnet.
If you just draw the lines continuously through the magnet, as on the right, then the lines look the wrong way round inside the magnet.

But you know that when you break a magnet, it becomes two magnets and you can repeat this many times until the magnet is a collection of magnetic crumbs.
Or, one explanation of magnets used to see the magnet (or any lump of ferromagnetic material) as made up of many domains, subdivisionsof the material which themselves had a permanent magnetism. In unmagnetised materials, they were randomly arranged (like top left), so there's no significant external field. When the material is magnetised, the domains align (as in the centre diagram) and give the external field.

Anyhow, that's what was in my mind when I tried to find a way to explain the field lines inside a magnet. I probably shouldn't have bothered!
Just don't draw lines inside at all, unless it helps you.

 

[rude man]

B field lines should be shown inside as well as outside the magnets. Pointing S to N inside the magnet and N to S outside.

 

[aspodkfpo]

No they don't exist - anymore than any of the lines do. You draw the lines that help you.

Sorry to confuse you. This was just me trying to think of a possible way to draw the lines through the magnet.
If you just draw the lines continuously through the magnet, as on the right, then the lines look the wrong way round inside the magnet.

But you know that when you break a magnet, it becomes two magnets and you can repeat this many times until the magnet is a collection of magnetic crumbs.
Or, one explanation of magnets used to see the magnet (or any lump of ferromagnetic material) as made up of many domains, subdivisionsof the material which themselves had a permanent magnetism. In unmagnetised materials, they were randomly arranged (like top left), so there's no significant external field. When the material is magnetised, the domains align (as in the centre diagram) and give the external field.

Anyhow, that's what was in my mind when I tried to find a way to explain the field lines inside a magnet. I probably shouldn't have bothered!
Just don't draw lines inside at all, unless it helps you.
View attachment 267927

Can someone show me a superposition diagram of the magnets inside of the magnetic bar? I have this model in my head where S N S results in a field towards N in the middle leading to no field going out of the magnet. Is the magnetic field from S N S N S N S N, where we ignore the things in the middle and just treat it as the leftmost S with a huge loop to the rightmost N?

 

[Merlin3189]

Can someone show me a superposition diagram of the magnets inside of the magnetic bar? I have this model in my head where S N S results in a field towards N in the middle leading to no field going out of the magnet. Is the magnetic field from S N S N S N S N, where we ignore the things in the middle and just treat it as the leftmost S with a huge loop to the rightmost N?

I hope I haven't driven you to chase red herrings! Your question here doesn't make a lot of sense to me.

SNS is not possible. Manetic poles always appear as NS dipoles. (I think there may be a theoretical monopole, but it's never been found. So don't look for that red herring.)

Another part of my diagram that is misleading (and it's not my diagram, just poached off the web) is the gap between the domains or mini magnets. They are really contiguous. Like putting two bar magnets together to make a single larger bar magnet. So yes to the SN SN SN SN SN that looks just like one big SN.

I think you're better off just following RudeMan's advice.

If you like, just think of magnets as if they were solenoids. They're made up of many coils, which, on their own would have a N and S pole. But together there are only poles at the ends. Inside where ever you stand, you just see flux lines coming towards you - like looking at a N pole- when you face one way, and see flux lines going away from you - like looking a a S pole- when you face in the other direction. These flux lines will be like the ones RudeMan describes and I drew (one of) in the right hand diagram.