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it's 3am and i'm just about ready to dielectric
just placing this down here so i remember to make a blog post on it
it's going to be great
did you know there's a thing called a Poynting vector sounds like a kabbalistic pun
any way yeah EM waves time later constantly refresh your browser
So imagine you have something that varies sinusoidally. You can write this as so after a sufficiently long time we just don't need to think about the free charge. That's a relief!
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The importance of a cause area, as ranked by 80,000 hours, my favourite effective altruist organisation, is calculated by per square metre. Unless it's useful (say, for largescale computation or energy gathering or something), civilisations will preferentially build small megastructures; this is expected, due to the squarecube law.
Large megastructures are still more efficient than the raw squarecube law would suggest; this would predict the log of the cost being half the log of the area. A plot is shown of the actual line of best fit, versus that expected by the square cub law. The explanation is that large megastructures can be much thinner compared to their size than small megastructures  a solid dyson sphere can be characterised as â€¦
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okay so functions are vectors.
and when you have vectors, you have basis vectors, which I'm gonna call e_{i}. these are the vectors that you can add together to make any other vector. in 2D space, they're called (1,0) and (0,1).
when you scalar product together two basis vectors, you get e_{i}.e_{j} = Î´_{ij}. that's a fancy way of saying that scalar producting a basis vector by itself gives you 1, and scalar producting a basis vector by a different one gives you 0. (0, 1).(0, 1) = 1. (0, 1).(1,0) = 0. nice.
now you can write down a function as a sum of your basis vectors. using braket notation, because I'm a fucking casual,
we switched to infinity because it turns out that if you change the variable and the limits at the same time it cancels out in this â€¦
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I'm passionate about circuit theory for some reason. I'm not sure why, but it's just really nice. It's like calculus, but in real life with little calculus machines.
At any point on the circuit, you have two associated quantities, which give you the state of the circuit: voltage and current.
Voltage is electrical potential energy. Batteries use their own energy to give the electrons in the circuit this potential energy. If the voltage drops, then this energy is released somehow; usually as heat.
This means that voltages around a closed loop must add up to 0. If you grab an electron, and then move it around the circuit, then it must lose as much energy as it gains. Otherwise, energy wouldn't be conserved.
Current is the flow of charges; it's prâ€¦
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This is circuit theory, which is physics, and therefore within the scope of this wiki. This is what I am going to tell my lawyer when you sue me.
These components follow Kirchoff's laws:
 Current flowing into a junction and current flowing out of a junction are equal (conservation of charge)
 Total voltage around a closed loop is zero (conservation of energy)
With these you can construct a differential equation that describes the behaviour of your circuit.
These are ideal components  nonideal components can be represented by adding resistors, inductors and capacitors to taste.
Worked examples coming maybe.
 Voltage source  Sets the voltage to . Associated with elastance and capacitance. Secretly just a gap.