If I have a series of magnets like below, do their fields combine to create one giant magnet, or do their individual fields remain?
Also, how do I calculate the force of attraction/repulsion between two magnets?
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2 Answers
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The magnetic field of a permanent magnet is the sum of the alignment of the magnetic dipoles of the electrons involved. (Since the electrons in the atom are also aligned with each other and in relation to the nucleus, the real summary magnetic field is smaller than the theoretically possible field if all subatomic particles were aligned in parallel). If you now hold a second magnet to the first one, you get a stronger magnetic field from the sum of the alignment of the magnetic dipoles of the subatomic particles from both magnets.
But be careful. There is a magnetic saturation for each material. This means that magnetic field lines will then emerge from the side of the bar magnet. An ever longer chain of bar magnets does not result in an ever stronger magnetic field at the poles.
On the other hand, there is also an amplification of the summary magnetic field. This happens when magnetic saturation has not yet been reached in one magnet and this magnet is connected to a second magnet. Then there is a self-amplifying effect in which the magnetic dipoles of the subatomic particles are more strongly redirected from their mutual dependence in the atomic compound and aligned in the direction of the common macroscopic magnetic field.
answered Aug 11, 2023 at 3:43
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$\begingroup$ So, the fields do combine as if the series was one magnet? $\endgroup$– YassinCommented Aug 11, 2023 at 8:52
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$\begingroup$ Quote from my answer „If you now hold a second magnet to the first one, you get a stronger magnetic field from the sum of the alignment of the magnetic dipoles of the subatomic particles from both magnets. But be careful. There is a magnetic saturation for each material. This means that magnetic field lines will then emerge from the side of the bar magnet. An ever longer chain of bar magnets does not result in an ever stronger magnetic field at the poles.“ $\endgroup$ Commented Aug 11, 2023 at 11:19
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$\begingroup$ Ok, so it does, until the saturation point? $\endgroup$– YassinCommented Aug 11, 2023 at 11:29
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The physical equations which describe magnetism are Maxwell's equations which are linear, and thus follow the superposition principle. The superposition principle states that the addition of two solutions of these equations is again a solution of the equations. If we apply this to your example and assume that the magnets don't influence each others fields (which isn't exactly true but poses a good approximation), we get that indeed the fields of the two magnets add up. So in simple terms: Yes, the fields of the two magnets "combine" to form an overall field which is created by the two magnets.
As for your second question: One way to do this would be to model each of your magnets as a magnetic dipole (which for the scope of your question should be a good enough approximation). Then you can calculate the force between the two magnets as \begin{equation} \mathbf{F} = \nabla (\mathbf{m_1} \cdot \mathbf{B_2}) \end{equation} where $\mathbf{m_1}$ is the magnetic moment of the first magnet and $\mathbf{B_2}$ is the magnetic field of the second magnet (I'm just considering two magnets for simplicity). The magnitude of $\mathbf{m_1}$ gives you the "strength" of the first magnet, whilst its direction corresponds to the orientation of the magnet. So for example, if you start by holding your magnets next to each other so that they attract each other and then you turn the first one by 180 degrees (which corresponds to a change of sign of $\mathbf{m_1}$) they will repel each other (which corresponds to the change of sign of $\mathbf{F}$ that follows from the change of sign from $\mathbf{m_1}$). The magnetic field of the second magnet can be calculated as
\begin{equation} \mathbf{B_2} (\mathbf{r}) = \frac{\mu_0}{4\pi} \left( \frac{3\mathbf{r}(\mathbf{m_2} \cdot \mathbf{r})}{r^5} - \frac{\mathbf{m_2}}{r^3} \right) \end{equation}
where now $\mathbf{m_2}$ is the magnetic moment of the second magnet. So in total we get
\begin{equation} \mathbf{F} = \frac{\mu_0}{4\pi} \nabla \left( \frac{3(\mathbf{m_1} \cdot \mathbf{r})(\mathbf{m_2} \cdot \mathbf{r})}{r^5} - \frac{\mathbf{m_1} \cdot \mathbf{m_2}}{r^3} \right) \end{equation}
Also this expression doesn't change, if we exchange $\mathbf{m_1}$ and $\mathbf{m_2}$ which just means it doesn't matter, which magnet you label as "first" magnet and which as "second". You can label them however you wish, it doesn't affect the outcome of the calculation.
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$\begingroup$ Just to clarify my first question, this means that the poles of the magnet would join so that there is a North on one end of the magnet chain and South on the other? $\endgroup$– YassinCommented Aug 10, 2023 at 14:58
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$\begingroup$ Yes, that is correct. So essentially, when you stick together two magnets, they form one bigger magnet with the north pole at one end and the south pole at the outer end. However, if you pull them apart, they will of course form two distinct magnets again. $\endgroup$ Commented Aug 10, 2023 at 15:42
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$\begingroup$ What happens to the fields if you put a series of magnets together like this... |N-S||S-N||N-S| The magnets are held so that they can't repel. $\endgroup$– YassinCommented Aug 10, 2023 at 16:14
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$\begingroup$ tutormyself.com/… that's a picture I found after a quick search that shows how the magnetic field looks in such a configuration (the magnets don't touch but qualitatively that doesn't change too much). However, notice that the configuration you described won't form a simple magnetic dipole anymore as (approximately) in the case of your original question! The field configuration is more complex now. $\endgroup$ Commented Aug 10, 2023 at 16:29