Chapter 4: Homework

Chapter 4: Chapter 4: Homework

Mataguchi’s monologue, POV

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"Maybe" I deadpan.

She didn’t reply immediately, her eyes seem to have scanned the room for a second before uttering. "Well I guess you can get lonely sitting alone in the empty room."

"..."

"How is student counseling?" She added walking to one of the president’s chair.

"..."

"I’m sure Oota hasn’t asked this but...what are you and Oota’s relationship?" She added sitting down with confidence, stroking her skirt from behind forward before sitting, her legs nicely pinned closed together to avoid been seen, well underneath there. Anyone can clearly see how self aware she was and slightly flustered but her consistent tsundere behaviour tries masking it all.

Cute

Wait this might be the second time I’ve called her cute since Elemetary.

"W-what? Why aren’t you saying anything?" She stuttered seeing that I was quite and haven’t spoken a word.

"Nothing, just brain dead as to why the two question you have just said fit into the same equation" I replied my eyes still on her.

"Look, I was just finding topics okay. I decided not to leave you here alone, you should at least be grateful you even have me here trying to ease your boredom" she protested her hands folded underneath her blazer.

"Oh and ’FYI’ I don’t actually care" she added.

Yeah right, she totally cares, if not she wouldn’t bother. Shion has this habit of ignoring what she doesn’t gets alone with or hates in general.

"Have you done your homework?" I asked

"Uh? What’s this about homework, of course I’ve done it. How long do you think the school break was?" She protested again.

"So you haven’t done it then" I added.

"....."

"....."

We both went into silence.

"Well..no, I was too busy with other stuff and I totally forgot about it" She finally responded honestly, her voice low barely audible to hear.

Hearing this and after observing her embarrassed expression I stood up.

"How many of the questions have you finished?" I asked while I walked up to her.

"12 subjects....i did those when I was bored so I-" she added but was distracted seeing as i had cut the distance between us.

"Eh? W-wait, what are you-" Shion stammered, her cheek crimson red. Her reaction was just too obvious well to me of course.

"What subject is left?" I asked walking passed her to the monitor screen behind her. Yeah I have to admit her reaction was something, if i were to put it simply ’cute’.

I tapped the monitor screen twice, with lightened the screen display to a pure white display.

I took out my phone out and shifted my eyes to her to hear her answer.

"...it physics, I have alright solved 29 of the questions, but then I got a invite to hang out with some friends so I kinda.... just decided to do it later but forgot about it totally.." her voice still remained low and but this time audible. Must have Bern embarrassing saying it out loud.

"I see so the last question then." I tapped on my phone, on my phone I located a folder ’hw’ sent it to the monitor screen. On the bottom side of the screen where my hands reached comfortably, I touched with a phone mini screen display with was like a smartphone, I listed the folder I had just sent. The mini screen also reflected on the main screen display, I the Physics questions where displayed, I scrolled down to the last for the last question.

Well I can see why she would forget to solve it, the question in question isn’t as subjectively hard to grasp than the other but it just doesn’t look solvable logically. And besides, she had solved 29 out of 30 correctly with no mistakes so letting go of one isn’t going to hurt her in any way.

Infinite grid of resistors...

The last question display on the board were as followed.

(SKIP TO NEXT Chapter IF UNNECESSARY OR CAN’T FOLLOW)

No.30

Question: What is the equivalent resistance between two adjacent nodes (or any two points) on an infinite two-dimensional grid of identical resistors?

That’s one brutal question though, well for the instructors at this school it might as well be a simple question for us.

"Are you ready, I will explain. I imagine you don’t need to write it down but memorise and get a grasp on it"

She’s only with her phone...

"O-oh yeah sure, I’m listening" Shion responded.

"You see, one cannot use standard series/parallel rules because the grid goes on forever." I explained

"Solving it requires advanced calculus, difference equations, or symmetry arguments such as injecting a current at one node"

"imagining it spreading symmetrically to infinity, and using the principle of superposition to find the answer."

"Solving the equivalent resistance Req between two adjacent nodes....let’s say A and B in an infinite grid of identical resistors is a classic problem"

If i Remember correctly, Oota finished his the day the questioned were given out... Well he did mentioned he doesn’t like taking homeworks home when we were hanging out that one time. The other two that he normally moved around with also finished theirs.

"Right.." Shion sigh, her eyes rolled as if tired of the lesson already. "So the most elegant technique relies on superposition combined with the grid’s symmetry to break the network into individual monopoles."

"Right" I approved.

"While we are act it I should just solve it on my phone. I just remembered something... my friends should be waiting for me" she added.

Right she did mention something about it some moments back.

She linked her phone screen to the monitor showing her progression. The screen displayed her solving progression so i could see.

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She broke it Step-by-Step with Superposition and Symmetry Solutions

For an infinite square grid where every resistor has a resistance of R0, the effective resistance between two adjacent nodes A and B is exactly Req = frac(R0/2)

"1,"

"Isolate the Current Source (Node A)."

Imagine injecting a total current I into Node A from an external source at infinity. Surely mist people understand where is going.

"Right, then i next i should apply Current Symmetry." shion said, already writing down the solution.

"Wait, You’re done?" I asked, honestly I know she was smart but I was expecting her to at least take a few seconds to note and process it mentally.

"I did took a few seconds you know, we you even watching?" Shion replied, her voice sounded a bit irritated for some reason. Maybe because of how I phrased the question.

"Because the grid extends infinitely and is perfectly isotropic, the injected current I spreads out symmetrically in all directions. At Node A, this current divides equally among the 4 connected resistors. Therefore, the current flowing away from Node A in each adjacent branch is frac1/4.

Shion had followed through perfectly without my explanation. All I did was watch.

3, Isolate the Current Sink (Node B). Imagine extracting the same total current I from Node B, sending it to infinity. By reversing the process, a current of frac(1/4) flows through each resistor towards Node B.

4, Superposition. When both operations happen simultaneously (current I enters A and exits B), the total current in the resistor directly between A and B is the sum of the currents from both scenarios.

With that Shion had concluded that the current flowing through the resistor between A and B was...

Shion wrote the numbers she had gotten on the board.

frac(1/4) + frac(1/4)= frac(1/2)

Next step was to Calculate Equivalent Resistance.

"Using Ohm’s Law, the voltage drop across the resistor between A and B is.." Shion muttered to herself before presenting progression.

VAB = (1/2)R0

Since the equivalent resistance is Req =(VAB/1)

She got Req =(1/2)R0/1= R0/2

Like that she was done with the first approach, immediately she got started on the others.

Mathematics approach

1, Difference Equations (Discrete Laplace’s Equation)

"The voltage at any node (x,y) in a uniform resistor grid must satisfy Kirchhoff’s Current Law (KCL), right? meaning the sum of currents at any given node equals zero." Shion explained writing some figures on the board through her phone.

(V(x+1,y) + V(x-1,y) + V(x,y+1) + V(x,y-1) - 4V(x,y) = 0

I listened without reacting in anyway, this was to test her own conclusions, if she stopped to seek validation from me than, it would be half merited. Shion already knows this wh8ch is why she didn’t bother, reacting to my lack of response. It might as well been a means of communication during situation like this.

Thoiugh what she had shown was a standard recurrence relation. To solve this, one typically sets a boundary condition

(e.g., the potential is zero at infinity) and uses the method of separation of variables (or \(Z\)-transforms), representing potentials as a Fourier series to find the exact node voltages.

Next was....

Advanced Calculus (Green’s Functions)

"For grids where we need the resistance between non-adjacent nodes, basic symmetry breaks down. I must rely on the discrete equivalent of electrostatics. the discrete Laplace equation right?." Shion uttered. For someone who seemed boredom minutes ago she was now fully attached to Solving the problem. She started writing on the topic.

In the continuum limit, an infinite grid approximates a uniform resistive sheet, and the discrete difference equation becomes the continuous.

She wrote down the formula on the screen.

2D Laplace equation V(x,y) = 0 before breaking it down.

1, A single node with injected current is treated as a point source. The potential at distance ’r’ behaves logarithmically

V(r) \propto ln(r)

2, Using Green’s functions (the mathematical impulse response of the system), one can integrate the response over the infinite lattice.

3, The potential difference between any two nodes is evaluated by mapping the grid coordinates to complex potentials, where the total resistance Rmn between node (0,0) and node (m,n) is extracted.

With that the sesson ended, honestly this was exhausting, I had to stand and watch her do all the work.

She homework aside, i had planned to ’help’, but the truth is I wasn’t interested on the result

A attribute of mine that had been with me since birth.

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"Isolation is far better than false warmth."

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