From df47aec31af547efd0ebc742a66197391776c781 Mon Sep 17 00:00:00 2001 From: Tyler Clarke Date: Sat, 5 Apr 2025 14:16:01 -0400 Subject: [PATCH] inductance hell --- site/default.html | 1 + site/posts/inductance-hell.html | 79 +++++++++++++++++++++++++++++++++ 2 files changed, 80 insertions(+) create mode 100644 site/posts/inductance-hell.html diff --git a/site/default.html b/site/default.html index 97e9ac5..5695563 100644 --- a/site/default.html +++ b/site/default.html @@ -36,6 +36,7 @@ [/]

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diff --git a/site/posts/inductance-hell.html b/site/posts/inductance-hell.html new file mode 100644 index 0000000..9c58471 --- /dev/null +++ b/site/posts/inductance-hell.html @@ -0,0 +1,79 @@ +[!] +[=post-] +

+ I intended to get a lot done today. I needed to write the multivariable 16.4 review, build my company webpage, solve the collatz conjecture and prove + that the real part of every nontrivial zero of the reimann zeta function is `frac 1 2`. And get my group to actually do some work on the project due in like + three days. +

+

+ I did not do those things. +

+

+ This week, physics was hellish. Specifically, one of the problems in the physics homework was hellish. I generally read the textbook regularly, + and attend lectures and pay attention and allat jazz, so I can usually finish the homework effortlessly. This week, however, we've just 'bout finished + Maxwell's equations, and inductance is... bad. I've missed family obligations, killed orphans, and had slightly less time than usual to play 0 a.d. (in order + of awfulness), purely for this one question. The really annoying thing is that it seems conceptually simple: we're given a bunch of parameters about a + solenoid and a resistive ring around it, and we need to find the induced current in the ring at a given time after a voltage source is applied + to the solenoid. Easy, right? The law of inductance should just about solve it. +

+

+ I started this nightmare with high hopes. The resistance in the ring is known, and we know that `V = frac {d phi} {dt}`, so by Ohm's law `I_{"ring"} = + frac {frac {d phi} {dt}} {R_{"ring"}}`. Because the b-field outside the solenoid is essentially negligible, flux in this case is just going to be + `A_{"sol"} * B_{"sol"}`, and `B_{"sol"}` is easy to find- it's just `mu_0 * frac N l * I_{"sol"}`. Taking the derivative with respect to `t` is also quite + simple: `frac {d phi} {dt} = A_{"sol"} * mu_0 * frac N l * frac {d I_{"sol"}} {dt}`. Not too pleasant, but not too hard either. +

+

+ It was about this point that Sally Ride came flaming down. +

+

+ See, the problem wants to know the induced current at some time very close to 0 but not equal to 0. Because current increases by an exponential function, this means + we're dealing with some nasty, nasty exponents. It took me embarrassingly long just to find the damn function to use: `I_"sol"(t) = I_"max" * (1 - e^{-t frac {R_"sol"} L})`. Ohshitohshitohshit + abort abort abort! What even is that! Where the hell did `L` come from! It turns out that the awful, awful `frac {R_"sol"} L` thingy is so pervasive it has a name: the time constant. + If you're familiar at all with exponential functions, this is obviously a damper on decay rate; the question is, how the hell do we actually find it? What is `L`? +

+

+ For whatever insane reason, `L` stands for "inductance". Inductance is the tendency of a given circuit to, well, induct: a measure of how the magnetic field produced by a changing current + inside the wire resists the current in the wire. This property is the actual reason current ramps up slowly; inductance only resists change in current. + I really, really, really don't want to work this out the hard way; fortunately, there's a ready-made equation for specifically a solenoid's inductance: + `L = frac {mu_0 N^2 A} {l}`, where `N` is the number of coils, `A` is the area of the solenoid's circular cross-section, and `l` is the length of the solenoid. + I don't want to calculate that (or even think about it, to be honest), but calling that `L` is just asking to confuse inductance with length, so I'm going to call it + something sensible: `L_"induc"`. +

+

+ Back to current. We need to find `frac {d I_{"sol"}} {dt}`, and for once this is straightforward. Expanded out, our formula for current is `I_"sol"(t) = I_"max" - I_"max"e^{-t frac {R_"sol"} {L_"induc"}}` + (`I_"max"` is just the current we'll achieve after a very long time: ohm's law tells us that it's `frac V {R_"sol"}`). The derivative of this is easy enough to take: + `frac {d I_{"sol"}} {dt} = -frac V R_"sol" e^{-t frac {R_"sol"} {L_"induc"}} * frac {-R_"sol"} {L_"induc"}`. We can cancel and simplify a bit to get + `frac {d I_{"sol"}} {dt} = frac V {L_"induc"} e^{-t frac {R_"sol"} {L_"induc"}}`. +

+

+ Now that we know the derivative of I with respect to time, we can smush it into our previous equation for magnetic flux, to get + `frac {d phi} {dt} = A_{"sol"} * mu_0 * frac N l * frac V {L_"induc"} e^{-t frac {R_"sol"} {L_"induc"}}`. Is it too late to change majors? +

+

+ But wait- it gets worse! This isn't actually the final equation we need. What we need is `I_{"ring"} = frac {frac {d phi} {dt}} {R_{"ring"}}`. + Substituting in `frac {d phi} {dt}` gives us `I_{"ring"} = frac {A_{"sol"} * mu_0 * frac N l * frac V {L_"induc"} e^{-t frac {R_"sol"} {L_"induc"}}} {R_{"ring"}}`. + Time to start filling in the blanks. We already know that `L_"induc" = frac {mu_0 N^2 A} {l}`, and the solenoid is known to have 1840 turns, a length of 0.1525 meters, + and an area of `pi * 0.0139^2`. I'll spare you the calculation: the result is `L_"induc" = 0.01693382781`. I don't like that number, Sam-I-Am, but at least it's a number + and not a disgusting derivative. Substituting in some more (voltage: 33.63 volts, time: 2.14 microseconds, `R_"sol"`: 153.7ohms, `R_"ring"`: 1637.1 ohms) gives us + `I_"ring" = 0.00001094959`. You know what I absolutely hate? That that answer is right. It has no business being right. That's a disgusting answer. +

+

+ I tried exactly those same steps with exactly those same equations a total of 6 times, having rederived them almost entirely every time. I had Gemma2 (locally) + calculate an answer, and when it failed, resorted to ChatGPT. I read discussion boards, and wiki pages, and LibreTexts, and even those shitty Hyper Physics thingies from GSU. + The only thing that worked was getting out my least favorite editor (surely you've moved to Lapce/Zed/Builder from the awful that is Code by now, right? If only any of them + had web editing worth a damn...), grinding out some templates, and writing down my thought process here in long and sarcastic form. Specifically, I rederived from scratch + equations that I've been trying and failing to find and apply for the last four days. +

+

+ I'm increasingly glad I didn't try to refund that domain name. +

+

+ There's still time in the day to do some things that I very much need to do, but I'll try to get Thomas 16.4 out tonight. 16.3 was awesome and 16.4 is looking to be a continuation of the awesome. + Until then, cheers, and happy Saturday! +

+[/] +[=title "Inductance Hell"] +[=author "Tyler Clarke"] +[=date "2025-4-5"] +[=subject "Physics"] +[#post.html]