81

Here's a map of the barometric pressure in the United States. The map contains isobars, which are lines of constant pressure. These are constructed by starting from an arbitrary point, and following the direction where the pressure doesn't change. Isobars don't "exist", in the sense that there isn't literally a big white line in the sky hovering over New ...


78

Individual iron filings will align their long dimension with the magnetic field. But then they will also feel the induced magnetization in other iron particles nearby, and they will tend to move toward each other till their points touch. This is what creates the strings of iron particles. When the needles cannot move from their site, one does not get lines.


78

The chemical bonds of the material keep it together. If the magnets you're thinking of are made of metal, then the chemical bond is the metallic bond, which is quite strong. You can get a sense of how strong it is if you try to rip a metal bar into two. Unless you are exceptionally strong, you probably won't manage – but you are probably able to pull a bar ...


61

why there are gaps between the the iron filling lines? Iron filings are ferromagnetic. They don't just show the field, they change it. ...hence the iron filings align themselves to stronger field lines. The filings self-organize into distinct lines because their presence concentrates the field. Magnetic field lines prefer to go through a ferromagnetic ...


34

The number of field lines is not a meaningful physical quantity, but only a useful tool to visualize the magnetic of electric fields. It is not a meaningful quantity because it is not measurable, for the reason that, as you said, "One can draw/imagine as many unique (curved/straight)lines as he/she wants in some specified finite area (assuming that each ...


32

My understanding is that field lines are just a visualization tool showing points of equipotential magnetic moment tangent to the line. Yes, field lines are just visualization tools we (humans) invented, they are not physical objects. I don't understand, intuitively, how an equipotential line could snap or break, or why that would result in a release of ...


31

Every knot is the boundary of an orientable surface. Such a surface is called a Seifert surface.$^\dagger$ For any given knot (with a given embedding in 3-d space), the flux is the same through two such surfaces. As usual, the flux can be calculated either by integrating $\mathbf{B}$ over the surface, or by integrating $\mathbf{A}$ around the knot. Figure 6 ...


25

Actually, we do! It's just that it's not the same "kind" of potential - and the reason for this is that magnetic forces work differently than electric forces. Magnetic fields, if you know, do not directly exert a force on charged particles, simply for being charged. (They would exert such on hypothetical "magnetically charged particles", but we've never ...


21

One way is through the Zeeman effect. The presence of a magnetic field in the region where the absorption line originates can split the energy levels of an atom into multiple components. Transitions between these split energy states then lead to absorption lines with multiple components at separate wavelengths. The number of components and the size of the ...


21

Consider the following bar magnet, with the unphysical field lines drawn around it. The real magnetic vector field is tangent to these lines and is represented by black triangles (a magnetic vector field always emanates from the north pole to end up at the south pole, though it continues inside the magnet): Now consider the following picture of two of ...


21

This paper by Lyutikov and McKinney describes this precise scenario. I'll provide a brief summary of their results here. One of the crucial underlying assumptions made when deriving the no-hair theorem is that the region outside the event horizon is a vacuum. However, a rapidly spinning, magnetized neutron star can induce vacuum breakdown, leading to the ...


20

Magnets are held together the same way all solids are held together: chemical bonds between the particles that make up the magnet, which are ultimately due to electromagnetism. Take two fridge magnets that attract each other, stick them together, and pull them apart. Then take an ordinary piece of metal, and pull on its ends with about the same force. This ...


18

You can reconstruct a vector field $\mathbf F$ from its divergence $\nabla \cdot \mathbf F$ and curl $\nabla \times\mathbf F$ and boundary conditions. More precisely, if $R$ is some bounded region on which $\mathbf F$ is twice continuously differentiable, then $$\mathbf F = \nabla \Phi + \nabla \times \mathbf G$$ where $$\Phi(\mathbf r) = \frac{1}{4\pi} \...


18

For a generic, oriented knot, you can construct an oriented surface which has the knot as it's boundary through the Seifert algorithm. Stokes' theorem says that the flux through any two such surfaces which share the same boundary must be the same. In principle, one could construct a Seifert surface for the trefoil knot, parameterize it, and then evaluate ...


15

To recover ${\bf B}$ from $\nabla\times {\bf B}$ and $\nabla\cdot {\bf B}=0$ you need to know $\nabla\times {\bf B}$ everywhere, not just at the point where you want to know ${\bf B}$


12

Rob Jeffries already has an excellent answer. I shall just add an image and some numbers to that answer. The spectrum in the presence of magnetic field will split certain absorption lines. The splitting $\Delta$ is proportional to the magnetic field strength. This splitting depends on the magnetic moment of the parent atom which is generally of the order of $...


11

The potential is a kind of primitive function of a vector field, primitive in the sense of being the reverse of a differentiation, ie., an integral with a variable upper limit. The derivatives of the potential in all directions represent the vector field; not all vector fields can be represented that way but some do, for example the electrostatic field. Some ...


11

This is indeed a perplexing and not fully-solved problem. It used to be stated that flux conservation alone might lead to fields of order $10^9$T which are deduced to be present at the surfaces of many rapidly rotating (and presumably young) pulsars. A 10 solar mass star on the main seqeunce has a radius of about 10 times that of the Sun, say $10^{10}$m. If ...


10

The linear system you've written, $$ \vec f= \begin{pmatrix} f_1 \\ f_2 \\ f_3 \end{pmatrix} =q \begin{pmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{pmatrix} \begin{pmatrix} B_1 \\ B_2 \\ B_3 \end{pmatrix} =q(\vec v\times) \vec B $$ is indeterminate, and it does not have a unique solution. You can easily ...


10

That does look like there are real lines, doesn't it? It's because each iron filing becomes its own magnet which affects the others. Notice how they're all crowded together close to the ends of the magnet, and then there's a region where they're thin, and then they get closer together farther away? I think that's because close up, the magnetic field of ...


10

We are indeed affected by magnetic forces as you say. They're just astonishingly weak. We can actually calculate them! Most of the time, our charge is so neutral that the story isn't any fun, so let's spice it up. Lets hook you up to a Van de Graaff generator, raising your potential to 100kV. That's enough to make a pretty darn impressive spark. To ...


9

There is a "magnetic potential" that appears in more advanced books and is defined as ${\bf B}= \nabla \phi$, just as ${\bf E}= -\nabla V$. (please excuse my equations if you are not familar with the $\nabla$ symbol) It's useful in regions where there is no current or no time-changing electric field, but it is "multivalued." Because of Ampere's law, if you ...


9

You are correct. But so is your teacher. If two magnetic fields are added at a point then the direction of the magnetic field at that point is given by the resultant, which is the same as the direction of the compass needle. Magnetic fields are vectors and there is always only one resultant no matter how many vectors are added together. Magnetic field ...


9

You are right: magnetic field lines can't snap or break because they are not physical objects. They are more analogous to elevation lines on a topographical map, or more precisely to lines perpendicular to elevation lines: to the fall lines on a ski slope. However, they do describe something physical, which is the magnetic field distribution. When the ...


9

I find it very confusing when people say that the strength of the magnetic field is proportional to no. of field lines/area. Why is this terminology still used? The terminology is still used because it is correct and it gives a graphical way to understand magnetic fields that is particularly intuitive to understand and apply. For an example, consider the ...


8

When you talk about the electric and magnetic fields interacting I would guess you are thinking of Maxwell's equations that give us the electric field $\mathbf E$ and magnetic field $\mathbf B$ for any arrangement of charges and currents. However it is not correct to say the electric and magnetic fields are interacting. Instead they are both components of ...


8

There are two main points of confusion in your reasoning: Any field can be decomposed as a sum of two fields: one with zero divergence and one with zero rotor. You say "any field can be constructed using its divergence and rotational" which is kind of obscure to me. If a field has zero divergence and zero rotor, it can still be finite! It does ...


8

The number of magnetic field lines you draw are just a means to describe the strength of a magnetic field, so it is a convention to draw more lines. If the field is stronger, but even if you draw only a few lines, the magnetic field between the lines still is there. These lines do not really exist. They just show in what direction the force is. You could ...


8

This is a very under-discussed topic in introductory physics classes. The instructor would just tell you this for a fact with no further discussion on the topic which is the source of a lot of confusion. Well, as answered already, it is a matter of convention. You "agree" to draw a certain number of field lines and then compare, "fixing" ...


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