# Are the field lines on a bar magnet diagram contour lines?

Following is the representation of magnetic field lines around a bar magnet.

The lines drawn, indicate the direction of magnetic field lines. Now I want to know that is this a contour mapping of the original space that is present around a magnet. I mean can we plot this in 3d by considering these lines as contour lines of the original 3d graph.

• These are not contour lines, these are field lines. It's a vector valued function's field lines Commented Aug 14, 2021 at 11:02
• I would say they're "dual" to contour lines. Commented Aug 15, 2021 at 4:40

No, the field lines are not contours.

Contours usually connect places of equal magnitude, height, for example, on a map of a hill.

The field lines on the diagram aren't connecting places of equal field strength, it's the gap between the lines that gives an indication of the field strength - the larger the gap between the lines, the weaker the field.

• Buy the contour works the same way, the gap between the lines gives the strength gap of a curve Commented Aug 14, 2021 at 11:02
• @Samyak Marathe, you could use the lines to do a 3D plot, as it's symmetrical around the axis of the magnet, but you'd have to use the gap between them if you wanted a 3D plot of the magnetic field strength Commented Aug 14, 2021 at 11:17
• In a way they are the opposite of contour lines. On a map you could draw lines that go up and down as quickly as possible (instead of as slowly as possible) and those would be the "gravity field lines" Commented Aug 14, 2021 at 19:59
• Would ‘contour lines’ be at 90º to the field lines? Commented Aug 14, 2021 at 20:45
• I felt like this answer didn't get to the core of the issue: that a non-conservative vector field (such as the magnetic field) cannot be depicted in terms of contours, so I added another answer. Commented Aug 14, 2021 at 21:12

No. Contour lines depict scalar fields, whereas the information in a magnetic field cannot generally be represented in terms of a scalar field.

Contour lines are a way of representing a scalar field (in the context of cartography vertical elevation $$h(x)$$). A magnetic field is not a scalar field, but a vector field, and so does not have contours in a straightforward sense.

That said: certain vector fields can be described in terms of a scalar field, for example $$\vec{v}(x) = \vec\nabla h(x)$$ is a vector field that contains the same information as $$h(x)$$, and this $$\vec{v}(x)$$ could be visually depicted using contour lines (if you so wished). The magnetic field however is generally not such a field: in mathematical parlance, we say it is not a conservative field. (As noted in the comments) A notable exception to this is a magnetic dipole, such as a bar magnet, whose magnetic field can be described in terms of a scalar field whose contours run perpendicular to the field lines. This, however, is a special case.

• The magnetic $\mathbf{H}$-field of a dipole magnet is in fact representable by a magnetic scalar potential $\psi$ as $\mathbf{H}=-\nabla\psi.$ You could make a contour plot for $\psi$ if you wanted to. The Wikipedia page has a nice image of such a contour plot (overlaid with an ordinary field line plot). This also applies to the restriction of the $\mathbf{B}$-field to the outside of the magnet because $\mathbf{B}\propto\mathbf{H}$ there.
– HTNW
Commented Aug 14, 2021 at 21:55
• @HTNW you are correct. thanks! In rehashing my answer I erroneously deleted the word "generally" which was meant to be my protection from just such a comment! I have amended my answer in light of your comment. I appreciate your point though that the scope of the bulk of my answer is more applicable to general plots of magnetic fields, rather than specifically the dipole. Commented Aug 15, 2021 at 0:34

No. But not because magnetic field lines are not contour lines - they kind of are.

In your case your 2d graphic is a 'slice' of a 3d case, so information is lost. If the magnet is a cylinder, then the 3d field could be radially symmetric about the rectangle given, and you could so reconstruct the 3d graph by rotation of the 2d 'slice' you have, and this would not be too unreasonable - but not actually correct.

What follows is an argument for why considering field lines to be a kind of contour is essentially not incorrect, along with some other details specific to electromagnetics in general.

You're on the right track with your intuition.

First note: The force which actually proceeds upon a charge from a B field always acts vectorily at a right angles to the direction of the B field (also to and in proportion to its velocity, the Lorentz force law). This is a strong 'hint' that considering B field to be a vector force field isn't quite right (at least, not as a 'real valued' force field).

Secondly: Magnetic monopoles do not, as far as is known, exist. There has never been direct experimental confirmation, and although technically one cannot prove a negative with a null result, in general engineers are quite happy to take it as a fact. The verified existence of a true magnetic monopole would change everything! Which is why it is often hunted for - but it has never been found, and may never be.

These two facts - allowing that you take them both as such - are enough to guarentee that magnetic field lines are contours of a sort: Always appearing only in closed loops for the same reason that contour lines of equal height on a 2d height map also always form closed loops.

But the true 'contour' here is actually in the H field, not the B field.

It's correctly pointed out that the B field line density is a reflection of field intensity - and this because 'field lines' are actually evocative of a phenomena that naturally forms when iron filings are placed nearby the magnet on a piece of paper.

The spacing which naturally forms (for given composition of sufficiently small iron filings, on paper close to ground on planet Earth, at least) has historically been used as a measurement of magnetic field intensity: As in, so many 'lines per square inch' which one can still find in the specifications of some patents.

It's about 64516 lines per square inch to a Tesla.

Even magnetic 'flux' is a mistaken concept - it talks about the apparent flow of something along the field lines, yet in fact there is no such movement (apart from the case of gyro-guided particles in a plasma, but they are not 'of' the field, just things being influenced by it, and furthermore their contribution to the field, both by magnetic current and charge density, is very often neglected without great error. And they're only 'guided' along the flux lines, they're actually moving along a helical trajectory, and it's the 'guiding centre' of the helix that moves along the field line, at least until a collision... In any case, that movement, by Lenz' law, is such that it tends to null out the fields).

Actually what happens is that energy propagates as electromagnetic waves, moving at all times perpendicularly to both the B and E field vectors. (See Poynting vector).

When you energize a solenoid coil, for example, the magnetic field propagates outwards (and inwards!) from the surface of the wires in the coil, and it will do so at right angles to the 'magnetic field lines'. And very quickly at the velocity described by the reciprocal of the geometric mean of the absolute local permitivity and permeability of the materials in which it moves. This is electromagnetic radiation - light, and it moves as fast. Quickest in vacuum, where both permeabilty and permitivity have their minimum values. In magnetic circuits, permeability is often many times higher, so propagation is correspondingly slower - and further hampered by Lenz's law when the field propagates into a conductor.

In the case of a superconductor, particularly a type-1 superconductor, it in fact essentially takes infinite time to penetrate! ( at least so long as the superconductor remains superconductive - push hard enough with a magnetic field, and it will abruptly break down. Cool the superconductor further below the critical temperature, and it will work against stronger magnetic fields, by a margin the grows the cooler you get it). This is because Lenz' law guarentees that a surface current will set up to perfectly null the introduced magnetic field, thus perfectly stopping the field from penetrating into the superconductor.

With a regular conductor, however, these 'eddy' fields soon die out, and the magnetic field easily penetrates. This is largely why it is very difficult to shield out magnetic interference. The best you can do is use high permeability material, and hope the field prefer to go 'around' along that material.

It's worth pointing out that when iron filings self-organize into the field lines, they are in fact changing the shape of the magnetic field - essentially 'sucking' it all into the lines so assembled. Some 'heat', in form of gently shaking the paper, is usually required to allow the filings to move. The process is best run like simulated annealing, where it works best starting with a lot of vibration, then easing it off slowly: hot then cold, but smoothly. This allows the filings a lot of movement at first, and then less as they settle down into lines.

This self-formation is an energy-minimising process, whereby the presence of the iron filings actually do change the field, due to allowing a higher permeability path for the magnetic "flux", which tends to cause the lines to space themselves naturally - the 'self assembling' part of the interaction.

This is because between existing 'lines', there is much less available flux / force - because the magnetic flux 'prefers' to take the 'easy' path along the higher-pemeability material of the filings.

So filings between 'lines' just tend to stay put at random, whereas filings able to partake in the magnetic path tend to be more strongly held there. This is why the 'simulated annealing' shake-down is needed - to give those 'lost' filings a chance to be close to a partially-assembled line.

What actually assembles the fillings is the energy released from the magnetic field when the flux can exist instead in the magnetisation in the material of the filings along the line. This results in a spatial gradient at right angles to the field line, acting to pull the filings towards the common assembly of a line.

The arrangment allows slightly less total energy to be stored in the magnetic field as a whole. Destroying the arrangement by pulling the filings away from the field, actually puts that energy back into the field. This happens generally whilst 'playing' with permanent magnets, and is a subtle part of why they at least seem 'permanent'.

You can also see this line self-formation interaction in three dimensions with surface tension of ferrofluid in the shapes it forms near poles of magnets. This also tends to 'pop' into a finite number of 'tendrils' depending on field strength - but also varying with variation of a variety of the fluid properties.

When it comes down to it, the depiction of 'field lines' and the discussion of them as existing entities is only really correct in the case of self-assembling iron filings, that are actually assembled.

The use of them as a measure is possible beause the self-assembly is reproducible (given the same magnet and filings, in general: the number of lines per area is a constant at least, but the specific positions of the lines each time will be more or less random - depending strongly on the distribution of where the filings happen to start out on the paper).

Depictions of magnetic field lines on graphs are also only visualization tools - generally of the B (flux intensity) field, but can be constructed of the H magnetomotive force fields as 'proper' contours (ie, of constant H-magnitude surfaces). Take a 2d 'slice' through such a 3d iso-surface map and you'll end up with something looking like 'lines of magnetic force', although equally arbitrary, as you can choose which values of 'equal' you want plotted. It is however, reproducible in a way that actual magnetic field lines won't be.

However, keep in mind for the general case (which can include effects from magnetisation, polarization, and the effects of time-varying electric currents and nearby plasma / moving charges), the relation between B and H (and E and D) is generally not simply linear, and includes other terms, which themselves may not be isotropic. The point being that B and H aren't just in proportion to each other, necessarily.

It's actually more common to see isosurface slice plots of the magnetic field magnitude - and this is at right-angles to the field lines, so iso-lines of on it are actually closer to paths that current would want to take to null the field, where any wire placed along them.