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Over the time i have been doing physics i have noticed a pattern related to energy that gets dissipated in systems which come to equilibrium after a certain time has passed. please hear me out with a few examples

(i) A capacitor connected to a a simple circuit

When we connect a capacitor to a circuit and close the switch of the circuit, charges flow, energy is redistributed. Let us say that the final charge on the intially uncharged capacitor is Q. Then since Q passess through an emf E(say !) it looses the energy equal to QE. On the other hand the energy stored in the capacitor is only 1/2QE. This means that another 1/2QE amount of energy has dissipated in the form of dissipative forces. We disregard the nature of this disspated energy, whether its light, sound, heat or any other form.

(ii)The transfer of water from a fully filled tank to another.

Consider the situation where two identical tanks are connected through a pipe, which has a valve. Initially the valve is closed and one of the tanks is fully filled upto the brim, while the other is empty.When the valve is opened finally after a long time the system reattains equilibrium. Since the pressure must same in both tanks therefore, the height of water column must be same in both of them, also volume is conserved. Let the initial height of the water column be H, area of the base of tank be A and the final height of water column be h then (hA +hA) = HA, this implies that h=H/2. Let us now analyse the energy. Initialy the energy was MgH/2 (since the COM is at H/2), finally the total energy is 2(M/2 x g x h/2) =MgH/4, again MgH/4 has dissipated away

There are many such examples that i have encountered. They vary from simple systems to quite complex unintuitive ones.They vary from different fields of physics. It happens in standing waves, rotational mechanics etc.

I know that this has to happen, because these systems must axiomatically come to equilibrium, and for that energy has to dissipate else the system will just oscillate with kinetic energy converting into potential energy and vice versa. What bamboozles me is that almost always the energy that has disspated is half of the initial energy that has been provided. Like i said this half stuff happens in standing waves also, and many more systems. I know its coming mathematically, But i am searching for a more intuitive ways to explain this, by leveraging symmetries, or making logical arguments. I need a feel of why this is happening.

EDIT

Take another very complex and non intuitive situation enter image description here

The solution is this: enter image description here You see even in such a complex which btw @Fakemod does not correspond to any direct linear integral so as to introduce a factor of 1/2, this total loss in dissipation is still 1/2, what do you say about this @Fakemod and @Starfall

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  • $\begingroup$ Again, I would argue that linearity is at work even in your complex example. Because $$\int \omega \mathrm d \omega=\frac{\omega^2}{2}+c\Rightarrow\frac 1 2 \mathrm d(\omega^2)=\omega\mathrm d \omega$$ Your angular velocity varies linearly with time. $\endgroup$
    – user258881
    Commented May 29, 2020 at 16:27
  • $\begingroup$ @FakeMod That is true but it was not visible directly or obviously, it required analysis, yet at the end it becomes linear, so why does nature prefer this linearity, that is my question, if you must frame it that way, i don't think this 1/2 factor ( a consequence of linearity) is a coincidence $\endgroup$
    – arnav009
    Commented May 30, 2020 at 10:31
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    $\begingroup$ No, it's definitely not a coincidence :-) And "why is it so?", would be a good new question which can be asked. To be honest, I don't think I know why. $\endgroup$
    – user258881
    Commented May 30, 2020 at 11:27

1 Answer 1

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You're wrong that the factor $ 1/2 $ in the capacitor energy implies any energy has been "dissipated away". This factor is simply a result of the fact that the voltage across a capacitor is not constant, but rather a function of the charge stored on the capacitor; so moving charges across a capacitor becomes harder over time. Since the defining relation of a capacitor is $ Q = CV $ with $ C $ a constant depending on the capacitor, and to move a charge $ dQ $ across the plates you do work $ V dQ $, you find that to store a charge $ Q_f $ in a capacitor, you have to do

$$ \int^{Q_f}_0 V dQ = \int_0^{Q_f} \frac{Q}{C} \, dQ = \frac{Q_f^2}{2C} $$

of work, and all of this work is then stored in the capacitor in the form of electrical potential energy, at least in this idealized setup.

Your second example is markedly different from the situation with the capacitor. Here, you explicitly violate the conservation of energy when you think about "when the system reaches equilibrium". Your system would explicitly not reach an equilibrium under conservation of energy; rather, its idealized dynamics are equivalent to that of a harmonic oscillator. In practice, of course, both such hydraulic systems and harmonic oscillators dissipate energy, but they do so through interactions that are not specified in the simple gravitational model.

It is true that under the effect of such external dissipative forces, your analysis of the equilibrium state of the system is correct; but the origin of the factor $ 1/2 $ in your example is simply that you chose to link two tanks together. If you had linked three tanks together, you'd get a factor $ 1/3 $, and if you had linked $ n $ tanks together you'd get a factor $ 1/n $. There's nothing deep about that.

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    $\begingroup$ Nice answer, however, it is worth stating the relation of factor of $1/2$ with the linearity of the expressed quantities ($V\propto C$, $E\propto h$,...). And thus upon integrating, we get a factor of half. It can also be equivalently explained by the linear nature of the graphs of the relevant parameters. Upon integrating this graph, we are basically finding the area of a triangle, and there comes our factor of half. $\endgroup$
    – user258881
    Commented May 29, 2020 at 10:35
  • $\begingroup$ @Starfall irrespective of the voltage changing at all times, a total charge Q has passed through the battery from a higher to a lower potential, all this energy must appear in some or the other form, whether it is a highly time variable work or a potential energy conservation of energy must be true, if you ignore dissipative forces where is all this energy which we account for through the energy stored in capacitor is going? Because then it is a direct violation of energy conservation, PS also check the edit $\endgroup$
    – arnav009
    Commented May 29, 2020 at 15:29
  • $\begingroup$ @FakeMod , I agree that in simple linear systems it can deduced that such a 1/2 term must appear but i have provided with a complex example in which such linearity is not as clearly visible, at least to me, please check the edits $\endgroup$
    – arnav009
    Commented May 29, 2020 at 15:31

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