Infinite Grid of Resistors
mathpages.com
mathpages.com
The electrical engineer suggests it's not measurable unless you apply current and also asks "when" after the current is applied referring to the distributed inductive and capacitive element and the speed of field propagation. The mathematician goes to a bar and has a stiff drink after hearing that.
That said there might be enough energy (infinity!) for anything to be possible.
If we assume this is an infinite grid in its own universe then nothing can actually move. The gravitational pull should be the same from every direction. If we assume the grid is perfect then there is no nucleation sites to start a collapse. The grid would be in perfect balance.
The same is thought about our universe. If there hadn't been small quantum fluctuations during the inflationary stage it would have taken much longer for what we see in the modern universe to form.
What it collapses into depends on the time for the bits to coalesce into. If it's a slow collapse you could get a mass large enough to form a neutron star (like thing) instead of black holes.
If those neutron stars crash into each other they can release a large amount of 'recycled' matter from all over the atomic spectrum back into this universe.
We impose initial conditions so that there is a congruence of motion of the connected resistors, so that we have a flavour of Born rigidity. Unlike in the special-relativistic Bell's spaceship model (in which the inertial motion of each spaceship identical save for a spatial translation), in our general-relativistic approach none of the line-of-connected-reistors elements' worldlines is inertial, and each worldline's proper acceleration points in a different direction but with the same magnitude. This gives us enough symmetry to grind out an expansion scalar similar to Raychaudhuri's, Θ = ∂_a v^a (<https://en.wikipedia.org/wiki/Raychaudhuri_equation#Mathemat...>). As an aid to understanding, we can rewrite this as 1/v \frac{d v}{d \tau}, and again in terms of a Hubble-like constant, 3H_0.
We can then understand Θ as a dark energy, and with Θ > 0 the infinitely connected line of ...-wire-resistor-wire-resistor-... is forced to expand and will eventually fragment. If Θ < 0, the line will collapse gravitationally.
> no nucleation sites
If Θ = 0 initially, we have a Jeans instability problem to solve. Any small perturbation will either break the infinite ...wire-resistor-wire-..., leading to an evolution comparable to Bell's spaceship: the fragments will grow more and more separated; or it will drive the gravitational collapse of the line. The only way around this is through excruciatingly finely balanced initial conditions that capture all the matter-matter interactions that give rise to fluctuations in density or internal pressure. It is those fluctuations which break the initial worldline congruence.
This is essentially the part of cosmology Einstein struggled with when trying to preserve a static universe.
In higher dimensions (2+1d, 3+1d) the evolution of rotation and shear (instead of just pressure and density) becomes important (indeed, we need an expansion tensor and take its trace, rather than use the expansion scalar above). A different sort of fragmentation becomes available, where some parts of an infinite plane or infinite volume of connected resistors can undergo an Oppenheimer-Snyder type of collapse (probably igniting nuclear fusion, so getting metal-rich stars in the process) and other parts separate; the Lemaître-Tolman-Bondi metric becomes interesting, although the formation of very heavy binaries early on probably mitigates against a Swiss-cheese cosmological model: too much gravitational radiation. The issue is that the chemistry is very different from the neutral-hydrogen domination at recombination during the formation of our own cosmic microwave background, but grossly a cosmos full of luminous filaments of quasi-galaxies and dim voids is a plausible outcome. (It'd be a fun cosmology to try to simulate numerically -- I guess it'd be bound to end up being highly multidisciplinary).
All of the mass does end up in the singularity, in finite time (at least for any finite subset of the black hole), but it doesn't automatically become super dense just because it's a black hole. It can remain quite ordinary for a very long time.
The kind of black hole that forms from a collapsing star is super dense. But a supermassive black hole forms differently, as a denser region of gas and stars during galaxy formation. The density can be lower than that of water. You could be inside it without even realizing anything is odd.
There is no known way to form black holes bigger than that. We had been discussing a pure thought exercise. Though it is possible that the universe as a whole has enough density to be a black hole. (Signs point to no, but it's an open question.)
I hadn't considered that sort of strange effect though! Makes me feel not so bad for 'never really getting it' because I just couldn't wrap my mind around the problem description's obvious inanity and the infinite edges.
It's simply an evaluation of your mathematical ability to manipulate the equations and overall understanding of them, wrapped up in a cute little thought experiment. This evaluation IS relevant to more realistic scenarios and therefore your grade and engineering ability.
But that's fine because knowing which path the electron took is not part of the problem. Both paths contributed to the resistance even if one was not taken.
We only have to worry about quantum effects if the probabilities are not a decent proxy for the partial-particles that we suspect don't exist. In this case, the Physicist can probably proceed directly to the bar and have a drink with the Mathematician.
A really difficult question: At each distance, what percentage of soldering errors in the grid can be tolerated before the fluke meter across the center square detects the fault? (That might actually be a thing as I've heard people talk about using changes of local resistance to detect remote cracks in conductive structures ... like maybe in a carbon fiber submarine hull.)
One is where the components on the schematic represent physical things, where the resistors have some inductance and some non-linearity, and some capacitance to the ground plane and so on. This is what we mean by schematics when we’re using OrCad or whatever.
There is another interpretation where resistors are ideal ohms law devices, the traces have no inductance or propagation delay or resistance. Where connecting a trace between both ends of a voltage source is akin to division by zero.
Sometimes you translate from the first interpretation to the second, adding explicit resistors and inductors and so on to model the real world behavior of traces etc. if you don’t, then maybe SPICE does for you.
Infinite resistor lattices exist only in the second interpretation.
Just wait infinite time for all the transient responses to die down. The grid to enter steady state and to became true to the schematic.
Mostly joking, but an idealized schematic requires idealized conditions to observe.
Going on something of a tangent: in engineering, it seems unusual to talk about “applying current,” it’s usually voltage (say, across a resistor) or some sort of an “electromotive force.”
I'll see myself out.
Why are the currents in the two node solution (not symmetric) a simple sum of the currents of two single node solutions (symmetric)?
Obviously the 2 node solution still has some symmetries but not the original ones that let us infer same current in every direction.
However note the problem is symmetrical about the vertical axis, so flip the figure. The current passing through the flipped paths should be the same as before the flip, so note down which i's equate to each other due to this.
Note that the problem is symmetrical about the horizontal axis, and do the same there. Note that the problem is symmetric when rotated 90 degrees, so do that. And so on.
In the end you'll have a bunch of i's that are equal, and you can group those into two distinct groups. Call those groups alpha and beta.
edit: Another way to look at it is that you can't use the available symmetry operations to take you from any of the alphas to a beta. This is unlike alpha to alpha, or beta to beta.
Another interesting aspect is that in an infinite grid, a spontaneous high voltage is going to exist somewhere at all times. It is probably very far away from you, but it's still weird.
The only type of person for whom intuition about infinity to form is not entirely unlikely are mathematicians.
Now, the same underlying theory also explains why there will always be diffraction from any finite boundary resulting in a reality which is indistinguishable from one where light actually takes all possible paths. But to argue that it actually does is arguably more meta-physics than physics. The demonstration is further compromised by the fact that the laser's diffraction performance is also presumably far from the physical limit.
So a cynic seeing that demo would say "Isn't that just due to some of the light from the laser being off-axis" -- and it absolutely is. The physics means that some of the light will always be off-axis, but the demonstration does nothing to establish that.
What do you mean by diffraction here? Are you talking about the bokeh?
The laser isn't interacting with any other elements than the air before it makes the gradient sheet light up. And the permisivity of the air is about the same as vacuum here.
Like, the grating is going to show a diffraction pattern similar to any pinhole aperture source [0] because of the 'half cancellations' they kinda explain, but not all that in depth.
But since there are no elements in the path yet, the only conclusion we can make is that of Feynman's - that the light is in fact taking all possible paths and then cancelling out to make the laser light we normally experience.
What am I missing? To me this demo is like mind boggling as it shows that the wave model is the correct one even with a laser.
[0] https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...
Something like (a) above, though I'm not confident that this is really showing that exactly.
Both l and A are infinite. So you get infinity/infinity, which is undefined, proving it's a silly problem and you should go do something useful with your time instead.
If you take an endless grid made of identical resistors and try to measure the resistance between two neighbouring points, the answer turns out to be about one-third of a resistor
[1] https://kirill-kryukov.com/electronics/resistor-network-solv...
Wouldn’t that be more efficient than converting current to heat?
They also just convert current to heat though. Some amount of current moving through a material with some amount of resistance always produces a fixed amount of current in accordance with Ohm's law. You can't really get away from that.
I guess the explanations always confuse me. Let’s say a short circuit with no resistors has a certain amount of power. Then we add a resistor and the power in the circuit goes down. The resistor isn’t turning the difference in power into heat, right.
Another useful heuristic is that heat is generated from what's left after all of of the other ways of converting energy are used up, such as light, chemical potential, and so forth. It's energy's last resort. The usefulness of a resistor lies in its simple voltage-current relationship, which is equivalent to saying that the only thing it generates is heat.
But let's talk about short-circuiting lithium batteries for example. They have a roughly 50 milliohm (aka 0.05 ohm) of "equivalent series resistance".
That means, if you short circuit a lithium battery with a superconductive wire (aka with 0 resistance), the circuit has a resistance of 0.05 ohms. We can compute the current with Ohm's law: I=V/R. V is typically 3.6 volts for li-ion batteries, R is 0.05 ohm, so I (aka current) is 3.6/0.05 = 72 amperes. 72 amperes * 3.6 volts is 259 watts. Now in the real world, the battery's chemistry would step in here and limit current in complicated ways, but this means that under the assumption that our battery would work as an ideal voltage source + a 0.05 ohm resistor, and if there was no extra heat coming from the chemical reactions, a shorted battery would produce 259 watts of heat.
We can add a 1 ohm resistor to the circuit, which means our circuit's combined resistance would be 1.05 ohm. Using Ohm's law again, we find that the current would be 3.6/1.05 = approx 3.43 amperes. 3.43 amperes * 3.6 volts is 12.35 watts of heat.
So thanks to our resistor, we're now producing 12.35 watts instead of 259 watts of heat, because the resistor limits the current going through the circuit. With a higher resistance resistor we'd produce even less heat.
A core idea here is that power consumptions equals heat. I don't understand the physics reasons why, but "this thing consumes 10 watts of power" means the same as "this thing produces 10 watts of heat". Higher resistance means less current which means less watts, which means both less heat and less power consumption because those are the same.
Look at it like this: resistors produce heat by definition.
What you are describing isn't a resistor. It's actually a switched power supply, but that material is probably more heterogeneous than you wanted.
There are ways to limit the current through some components regardless of the applied voltage (to a sane level) without producing much heat. These are active switching mode DC-DC converters widely used eg. for driving LEDs in (higher quality) light bulbs and charging batteries.
A voltage drop with a flowing current is power (P=VI).
There is literally nothing you can do to avoid resistors dissipating that power as heat, it's just what they are. If they didn't do it, they wouldn't be resistors.
What you can do is use larger resistances which need less current to see the same voltage (e.g. change a pull up from 10k to 100k or higher, but that's more sensitive to noise), or smaller resistances that drop less power from a given current (e.g. a miiliohm-range current shunt, and then you need a more sensitive input circuit) or find another way to do what you want (e.g. a switched-mode power supply is far more efficient than a voltage divider at stepping down voltage). This is usually much more complex and often requires fiddly active control, but is worth it in power-constrained applications, and with modern integrated technology, there's often a chip that does what you need "magically" for not much money.
Possibly based on this ranking. Everything sub-mathematician is 2 points? Maybe there's subdivision of points.
Here, I don't think it's even useful to look at this problem in electronic terms. It's a pure math puzzle centered around an "infinite grid of linear A=B/C equations". Not the puzzle I ever felt the need to know the answer to, but I certainly don't judge others for geeking out about it.
How else to create students capable of solving problems we cannot anticipate today?
Not to mention, that understanding strange problems is a very efficient way to broaden horizons.
It's a weird butterfly effect moment in my career though. I had an awesome professor for circuits 1, and ended up switching majors to EE after that. Then got two more degrees on top of the bachelor's
Last I checked, they don't. I certainly never hit an "infinite grid of resistors" in general circuits and systems except as some weird "bonus" problem in the textbook.
Occasionally, I would hit something like this when we would be talking about "transmission lines" to make life easier, not harder. ("Why can we approximate an infinite grid of inductors and capacitors to look like a resistor?")
It's possible that infinite grid/infinite cube might have some pedagogical context when talking about fields and antennas, but I don't remember any.
The people who loved application and practical solutions went to industry, the people who got off spending a weekend grinding a theoretical infinite resistor grid solution went into academia.
A lot of STEM education is more along the lines of "take the rapid-fire calculus class, memorize a bunch of formulas, and then use them to find the transfer function of this weird circuit". It's not entirely useless, but it doesn't make you love the theory.
Hey now, those actually come up sometimes.
I always thought this problem was a funny choice for the comic, because it’s not that esoteric! It’s equivalent to asking about a 2d simple random walk on a lattice, which is fairly common. And in general the electrical network <-> random walk correspondence is a useful perspective too