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Veritasium "How Electricity Actually Works"
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electrodacus:

--- Quote from: hamster_nz on May 22, 2022, 04:58:24 am --- :-// Oh come on, you know the formula: work = force x distance, 

If you or I push down on a lever, lifting a weight, it is still work, even though the weight on the far end can move the lever back to the original position, undoing our efforts.

--- End quote ---

You will be storing potential energy.
So say you put in 144mWs by dropping a weight from some height and it is connected through a pulley to another weight that is lifted (this will be the input).
Say lifting that other weight gained you 72mWs in potential gravitational energy and 72mWs were wasted as heat due to lifting mechanism friction.
Now all you have lost is the 72mWs due to friction the other 72mWs are still available in the same form as the original so you can say you moved potential gravitational energy from one weight to another not very efficiency just 50% efficiency.

I guess the above is a decently accurate analogy to what happens with the capacitor circuit.
Best case ideal case when you have no friction you are able to lift the same weight as the one providing the input to the exact same height.
So will you say that you did some work or did you just transferred the potential kinetic energy from one weight to another ?
hamster_nz:
Here is the mechanical analogy to the two capacitor problem that actually fits.

Springs can store energy, Stored energy is proportional to the 'stretch' squared - just like a how capacitors store energy. In this analogy, the spring is our capacitor.

Flywheels can store kinetic energy. They also have very good bearings that cause minimal energy loss. This is our deliberately introduced inductance.

Ropes can transfer the forces (admittedly only when it tension, but we will make sure they stay in tension). Here I am making no statement if these are the wires, or the electric field. These are just a mechanical way to transfer the forces.  When the rope is moving it is assumed to have minimal momentum. Consider the momentum of the rope as self-inductance - if the flywheel is very light enough (or things are arranged so no flywheel is needed), this might becomes important in in how the system behaves. But with any sensible flywheel it becomes unimportant.

The flywheel has a clamp on it. When the clamp is on the rope can't move, so no force can be transferred through a clamped flywheel. This is our switch.

The setup
See the lefthand side of the attached image.

We have two springs bolted to the ground, and a very heavy flywheel attached to the roof. We tie a rope to the lefthand spring, wrap it around the flywheel a couple of times so it can't slip, and tie it to the righthand spring, so it is nice and snug, but with minimal stretch on either spring. We check that the springs are both at equal height, and then clamp the flywheel in place.

This is our "zero energy in either capacitor" state.

Charging
With the flywheel clamped, we shorten the righthand rope till we get one unit of stretch in it. We tie the rope off, and trim it off nice and neat.

All of the energy is now in the righthand spring and the apparatus is armed and the experiment can proceed.

The experiment.

We release the clamp. The extra tension of the right hand side causes the flywheel to spin up in the clockwise direction, and the left-hand spring start to stretch.

What I expect to see, in an 'idea' world
When the springs get to equal length, in a perfect world half the energy in the flywheel, so it keeps the rope moving, slowly spinning down, finally coming to a stop when the LH spring now has ALL the energy in it. The cycle reverses, and continues on forever - i.e. the flywheel start turning counterclockwise, and ALL the energy is transferred back to the RH spring.

What I expect to see, in the real world
Energy gets lost due to friction and other effects (like rope slippage and stretch). Regardless of the weight and quality of the flywheel, springs and rope the system will settle down with only half the initial energy stored, shared equally between both springs. The only thing that is different is how long it takes to settle down.

That is it. there is no more to the "Two capacitor problem" than that.
snarkysparky:
Electrodacus has agreed that there is power available to do work in the return circuit. And that this current flowed in and out of  the capacitor terminals.   As to whether this power went "through" the capacitor i guess we will never agree. 

Am I correct ?
PlainName:
That appears to be the gist of it.
electrodacus:

--- Quote from: hamster_nz on May 22, 2022, 09:19:16 am ---Here is the mechanical analogy to the two capacitor problem that actually fits.

Springs can store energy, Stored energy is proportional to the 'stretch' squared - just like a how capacitors store energy. In this analogy, the spring is our capacitor.

Flywheels can store kinetic energy. They also have very good bearings that cause minimal energy loss. This is our deliberately introduced inductance.

Ropes can transfer the forces (admittedly only when it tension, but we will make sure they stay in tension). Here I am making no statement if these are the wires, or the electric field. These are just a mechanical way to transfer the forces.  When the rope is moving it is assumed to have minimal momentum. Consider the momentum of the rope as self-inductance - if the flywheel is very light enough (or things are arranged so no flywheel is needed), this might becomes important in in how the system behaves. But with any sensible flywheel it becomes unimportant.

The flywheel has a clamp on it. When the clamp is on the rope can't move, so no force can be transferred through a clamped flywheel. This is our switch.


--- End quote ---

You got to complicated and the analogy is wrong.
I guess you wanted to discuss the two parallel capacitor problem but more recently we are discussing a ideal voltage source and a 1000uF capacitor charged by that.
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