Yes, I transposed the equation in my head incorrectly from C = Q/V. Thanks for the catch.Small typo here. It's actually Q = C*V. But your result is correct, of course.
Yes, I transposed the equation in my head incorrectly from C = Q/V. Thanks for the catch.Small typo here. It's actually Q = C*V. But your result is correct, of course.
You did not address my comments in post#9 concerning this situation.the Electrician
What you describe is ohmic losses, which were postulated to be zero.
I can't access the first paper but this is the conclusion of the second:What you describe is ohmic losses, which were postulated to be zero. Current thinking is that radiation accounts for the loss of energy; here are the calculations:
http://kirchhoff.weebly.com/uploads/1/6/3/0/1630371/t.choy_ajp72-662.pdf
If we step the potential causing the electrons in the cross-section of the conductor to move so fast as a group that there aren't collisions (reducing PE by conversion to thermal energy) with other electrons in the conductor (low electrical resistance) this means they will hit the capacitor plate first and the potential energy of total electron flow can become a significant factor of total energy like we see in a high energy beam in vacuum. This PE will be converted to some type of KE when it hits the plates.VIII. CONCLUSION
We have extended the discussion of the radiation from the
transient switching of charges between two capacitors. We
have shown that the capacitors themselves can radiate, using
a point electric dipole model. We found this radiation to be
small but not insignificant,
..
The calculation of the radiation
would be a good exercise for an undergraduate student using
the methods developed here. Our results show that although
the details of the capacitor radiators are unimportant for the
recovery of the missing energy, they are important for the
study of the transient response and electromagnetic compat-
ibility.
...
The analysis shows again that
the two-capacitor problem with radiation still remains elu-
sive
Sorry. I got the wrong link in there. Try this:I can't access the first paper...
I didn't know you wanted a response. It is already fairly well known that the energy loss occurs whether we postulate some resistance in the circuit or not. That is, even if R is zero, the same energy loss occurs:You did not address my comments in post#9 concerning this situation.
Further what happens if you take the limit as R tends to zero in my post #15 analysis?
This appears not to be so. The papers I referenced show that EM radiation can account for the loss. Here's another reference that doesn't include the full paper:Unless the potential is so great that we see accelerations close to a large fraction of light speed the losses from EM radiation will be small.
I think that's the key here. The "paradox" only exists because of an apparent failure of the mathematics. Of course the math doesn't really fail, and using the right math will resolve the paradox.Further what happens if you take the limit as R tends to zero in my post #15 analysis?
All this is like sayingIf the capacitors are made of superconductor material, as are the connecting wires of finite length, when the connection is made an oscillating current will exist forever. Surely EM radiation (not thermal radiation) will eventually carry away the energy.
I explained how idealization leads to the "paradox" in the first 4 paragraphs of post #1, and how real world imperfections allow the problem to be solved. You've reiterated and given further examples, but it didn't seem to call for a response from me.Hello, Electrician,
My comment in post=9 concerned overidealisation.
I am basically suggesting that just as trying to solve a beam, built in at both ends by using the three equations of static equilibrium (Resolving vertically, horizontally and taking moments) will fail,
So will trying to solve the two capacitor problem by excessive idealisation of circuit elements and the use of KVL/KCL or equivalent circuit theorems.
In the case of the beam the real world elastic properties of the beam need to be invoked, in the case of the two capacitor circuit, the real world 'imperfections' of capacitors and wires are needed.
Of course. I can't help but agree--if the ordinary resistance of the wires and material of the capacitors are made zero, as with superconductors, some other means will operate. It seems to me that it goes without saying. Some people find it interesting to explore what those means might be. A google search finds many papers on the topic, and there doesn't seem to be a consensus yet. When the connecting wires are of finite extent (with finite inductance and non-zero radiation resistance), I find the radiation hypothesis appealing.All this is like saying
If my beam is too narrow, like the Tay bridge, it will twist and topple over,
If it is to flexible, like the Tacoma Narrows bridge, it will wobble into failure,
and so on
That is the lost energy will be dissipated by whatever means available to it in the real world.
The interesting or paradoxical part of it is that the loss is predicted with great precision by a formula, and the result is exactly the same no matter what the mechanism is. It's hard to imagine that aggressively eliminating losses by, for instance, using short, superconducting conductors, is all predicted to have exactly zero impact.Some people find it interesting to explore what those means might be.
I'm adding to, not contradicting, anything you are saying here.Of course. I can't help but agree--if the ordinary resistance of the wires and material of the capacitors are made zero, as with superconductors, some other means will operate. It seems to me that it goes without saying. Some people find it interesting to explore what those means might be. A google search finds many papers on the topic, and there doesn't seem to be a consensus yet. When the connecting wires are of finite extent (with finite inductance and non-zero radiation resistance), I find the radiation hypothesis appealing.
I think it's called the second law of thermodynamics.A similar mechanism operates in electrical engineering with the disposition of currents (real or complex) between available paths, although I don't know of a name for this.
by Aaron Carman
by Aaron Carman
by Aaron Carman
by Don Wilcher