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.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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
Hey now, those actually come up sometimes.
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.
