# One capacitor charging another capacitor

## Homework Statement

You have one capacitor C1 (1uF) charged to 10V.
Now the capacitor is switched to charge another capacitor C2 (0.25uF),
Whats the voltage on C2?
Everything's ideal. No loss caps, no wire resistance.

## The Attempt at a Solution

I tried,
Q1=C1V1
Q2=C2V2

total charge will be constant, so Qtotal, Qt=CtV
Both caps are in series, so Qt= (C1+C2)V

(C1+C2)V = C1V1+C2V2

V2 is 0v

V = C1V1/(C1+C2)

So, V=8V.

But when I do E = 0.5*C*V^2 for C1. (80uJ)
and then for both the caps(32uJ and 8uJ), the numbers don't add up.

## The Attempt at a Solution

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gneill
Mentor
Was there a question to go with this?

Sorry, the question is to find the voltage at capacitor C2.

gneill
Mentor
Sorry, the question is to find the voltage at capacitor C2.
It looks like you found it when you calculated the 8V.

You could also have found it by interpreting the capacitors as being in parallel, and placing the total charge on the equivalent capacitor C1+C2.

Don't fret about the loss of energy between before and after; When the charges were redistributed they had to move. Anytime something moves or changes spontaneously in nature (that is without some external energy source to drive it) it's because the system is seeking a lower energy state and one is available to be filled. If the components were truly ideal, with zero resistance, then the charge would continue sloshing back and forth between the capacitors (even a perfectly straight, 1cm length of wire has some inductance!). In real life the oscillations would eventually bleed away the energy via electromagnetic radiation and the system would settle down to the quiescent steady state.

rude man
Homework Helper
Gold Member
But when I do E = 0.5*C*V^2 for C1. (80uJ)
and then for both the caps(32uJ and 8uJ), the numbers don't add up.

Your capacitor leads have finite resistance. If you connect C1 to C2 with a resistor, no matter how small, you will find by the usual analysis that the missing energy is wholly disspiated by that resistor.

NascentOxygen
Staff Emeritus
Your capacitor leads have finite resistance. If you connect C1 to C2 with a resistor, no matter how small, you will find by the usual analysis that the missing energy is wholly disspiated by that resistor.
And in the dielectric losses. (unless the dielectric is vacuum)

I know the losses- ESR, dielectric loss, wire resistance.
The Q=CV equation does not contain a loss term, but the result contains loss of energy. How is this possible?
Let's assume ideal capacitor, idea wire (zero resistance).

rude man
Homework Helper
Gold Member
I know the losses- ESR, dielectric loss, wire resistance.
The Q=CV equation does not contain a loss term, but the result contains loss of energy. How is this possible?
Let's assume ideal capacitor, idea wire (zero resistance).
You have to accept that there is SOME resistance. It can be as small as you like, but not zero. ∫i^2*Rdt from 0 to ∞ will always be the difference between energies before & after connection is made, for ANY value of R.

This is a well-known conundrum. It is always dangerous to deal with singularities. So just accept that R cannot be absolute zero.

cmb
You have one capacitor C1 (1uF) charged to 10V.
Now the capacitor is switched to charge another capacitor C2 (0.25uF),
Whats the voltage on C2?
Everything's ideal. No loss caps, no wire resistance.
I would say that the reason you are getting caught in a quandry over how to answer this is because it cannot be answered. The question is, itself, 'wrong'.

This, as posed, would lead to the condition well described in #4 above:

If the components were truly ideal, with zero resistance, then the charge would continue sloshing back and forth between the capacitors
That is to say, there would be no static value of voltage on C2 that could be defined. If everything was perfect and zero resistance with no inductance, the frequency of VACC2 would be infinite!

Think of it as a Newton's cradle with two balls of different masses. The little one is stationary and the big one is dropped. There are no losses. How high is the smaller ball? You see, it is a time-dependent thing and the energy just swings one into the other, then back again.

To answer the question, you must pragmatically include an assumption about energy losses, else the answers you get are meaninless. If you like, you could work out the max and min volts on each capacitor during you theoretically perfect oscillation, because those would be finite values. I'll start you off; the max volts on C1 is 10V, and the min on C2 is 0V. ;)

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NascentOxygen
Staff Emeritus
The smaller the resistance, the greater the current surge, therefore the higher the i2R losses. With no wire resistance, the L-C ringing and associated radiated electromagnetic energy (radio waves) would be the principal energy loss from the system.

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NascentOxygen
Staff Emeritus
That is to say, there would be no static value of voltage on C2 that could be defined. If everything was perfect and zero resistance with no inductance, the frequency of VACC2 would be infinite!
If there was no inductance, then what mechanism would cause the charge redistribution to overshoot?

rude man
Homework Helper
Gold Member
The smaller the resistance, the greater the current surge, therefore the higher the i2R losses. With no wire resistance, the L-C ringing would radiate electromagnetic energy (radio waves), this itself posing an energy loss from the system.
Incorrect. The smaller the resistance, the greater the current surge, but the shorter the duration of the surge. So the integral of i^2*R is the same over t=0 to ∞.

Work the math and forget radio waves.

NascentOxygen
Staff Emeritus
We can't simply "forget" EM radiation when in the limiting case it is the only existing loss mechanism.

cmb
If there was no inductance, then what mechanism would cause the charge redistribution to overshoot?

But we should still be able to theorise about idealised systems, and to address your question you should think again on the Newton's cradle example. You are asking why the falling ball does not instantly stop, but its energy 'overshoots' into the other smaller ball. (the analogy is - ball height is voltage).

There is no 'overshoot', there is merely the bouncing of energy back and forth. There are two competing, opposed, osillatory drivers (as there usually are with oscillators!): You have the capacitors trying to even out the charge such that the smaller capacitor takes proportionately more voltage, whilst you have the system trying to even out the voltage.

Nothing complicated in this, just think about the two germane elementary rules in electronics; the potential across serial capacitors will distribute itself proportionately to the capacitance, yet a differential voltage will cause a current.

cmb
We can't simply "forget" EM radiation when in the limiting case it is the only existing loss mechanism.
Yes we can. It is a theoretical device, and it is infinitely small. No radiating elements.

NascentOxygen
Staff Emeritus
There is no 'overshoot', there is merely the bouncing of energy back and forth.
If there is no overshoot, there is no oscillation--there can be no "bouncing back and forth".

cmb
If there is no overshoot, there is no oscillation--there can be no "bouncing back and forth".
I am not describing inductive overshoot, per se.

As I stated, this is not a real-world example for which there would always be some inductance. An infinite current (I) and a zero inductance (L)... what magnetic energy does the inductance formula [0.5xLxI^2] give you with those two values?

Don't stress over it... the question is answered. It is either unrealistic question, or you make an assumption of some teeny amount of ESR so it damps promptly.

NascentOxygen
Staff Emeritus
I am not describing inductive overshoot, per se.
Then what back and forth phenomenon are you talking about?
As I stated, this is not a real-world example for which there would always be some inductance. An infinite current (I) and a zero inductance (L)... what magnetic energy does the inductance formula [0.5xLxI^2] give you with those two values?
If there is no inductance, there can be no stored magnetic energy. That's not too complicated, is it?
or you make an assumption of some teeny amount of ESR so it damps promptly.
Damps? What exactly do you imagine we need to damp? With no inductance, it is not a second order system, there is oscillation, so there is nothing to dampen.

rude man
Homework Helper
Gold Member
Yes we can. It is a theoretical device, and it is infinitely small. No radiating elements.
Good point! No antenna! Hertz would have understood.

cmb
If there is no inductance, there can be no stored magnetic energy. That's not too complicated, is it?
It is complicated by the fact that it is an impossible question. You say there is no magnetic energy because there is no inductance. But there is an infinite amount of current. In fact, there is an infinite amount of current squared!! I say my infinite current [squared] trumps your zero inductance! There WOULD BE magnetic energy there, in this dream-world fantasy!!!

This is quite a simple problem to sort out, even if you do not have all the info for the losses. All you need to recognise is that there will be energy losses. Then you treat the two capacitors as parallel (hence you can deduce a sum capacitance, C1+C2). Then you calculate the amount of charge present before the switch was thrown. The charge cannot be lost, as the + and - terminals of each capacitor, so to speak, are mutually isolated. So we then know charge, and total capacitance - from whence the voltage is defined!

Interestingly, you could stick a megohm or a milliohm resistance in the way, a Henry's worth of inductance, with a big Q or a little Q, or none at all, but the energy lost would be the same. Intuitively a bit wierd, but there it is!! The charge and capacitances are immutable values, so the final energy must also be immutable.

(It is so counter-intuitive, I am not even sure of this!! Can anyone spot any flaws?)

NascentOxygen
Staff Emeritus
It is complicated by the fact that it is an impossible question. You say there is no magnetic energy because there is no inductance. But there is an infinite amount of current. In fact, there is an infinite amount of current squared!! I say my infinite current [squared] trumps your zero inductance! There WOULD BE magnetic energy there, in this dream-world fantasy!!!

This is quite a simple problem to sort out, even if you do not have all the info for the losses. All you need to recognise is that there will be energy losses. Then you treat the two capacitors as parallel (hence you can deduce a sum capacitance, C1+C2). Then you calculate the amount of charge present before the switch was thrown. The charge cannot be lost, as the + and - terminals of each capacitor, so to speak, are mutually isolated. So we then know charge, and total capacitance - from whence the voltage is defined!

Interestingly, you could stick a megohm or a milliohm resistance in the way, a Henry's worth of inductance, with a big Q or a little Q, or none at all, but the energy lost would be the same. Intuitively a bit wierd, but there it is!! The charge and capacitances are immutable values, so the final energy must also be immutable.

(It is so counter-intuitive, I am not even sure of this!! Can anyone spot any flaws?)
So, that's a summary of the situation. But your back and forth oscillation? We're waiting to hear how you account for that, since you say your explanation doesn't involve inductance.

cmb
..your back and forth oscillation? We're waiting to hear how you account for that..
(Sigh!)

I believe you are the only one waiting to hear. And there is nothing to hear. The question itself is a theoretical proposition, fraught with inconsistencies. The point was made simply to say that if there were NO energy losses, then the 'rest' condition (as per my definition above) would only be transitory as there would still be this other energy in the system. Electrostatic, magnetic, I don't care!?! :grumpy: Whatever that energy is, it would push the system away from that 'rest' condition. Then it would try to come back to that rest condition. Then it would move away. Then it would try to come back... etc.. It would never reach an end. But my posts here have!....

NascentOxygen
Staff Emeritus
cmb said:
Yes we can. It is a theoretical device, and it is infinitely small. No radiating elements.
Good point! No antenna! Hertz would have understood.
I have my doubts that Herr Hertz would have understood, actually. But since you at least seem to understand perhaps you can clear up this conundrum for the rest of us.

How are you going to pack a bunch electrons (remembering that 1 coulomb =6.24×10^18 electrons) into something infinitely small, when, for starters, we know that individual electrons have finite size?

Just wondering .....

gneill
Mentor
How are you going to pack a bunch electrons (remembering that 1 coulomb =6.24×10^18 electrons) into something infinitely small, when, for starters, we know that individual electrons have finite size?

Just wondering .....
Actually, as far as we can tell electrons are point particles -- no size! But what will kick in to prevent such close packing is the Pauli Exclusion Principle. You just can't superimpose all those electrons in the same place with the same quantum state without a really, really good pair of vice-grips! Or the gravitation of a black hole...

Anyways, you can only throw out so many laws of physics for a gedanken experiment before you're no longer discussing physics or anything remotely like it. Resistance you can safely do away with since we know that superconductors exist. But you cannot wave your hands and eliminate inductance any more than you can banish inertia in a kinematics problem and still have it make sense. And accelerated charges WILL radiate.

When current flows from one capacitor to another energy can be lost in a variety of ways
1) due to resistance in connecting wires.... this can be made zero
2) sparking at the switch.... this can be made zero
3) electro-magnetic radiation (could be radio waves) from the connecting wires. Whenever a current changes electro-magnetic radiation is produced. (That is what a radio transmitting aerial is). Current is changing during the transfer of charge from one capacitor to another... or from a battery to a capacitor. So there will always be some electro-magnetic radiation.... it can be detected on a radio set as a crackling 'interference'
In mechanics there is an analogy with placing a weight on a spring, if the weight is lowered onto the spring then the spring achieves its equilibrium position and the energy stored is 0.5F x extension. If the weight is released then the energy given up by the weight is F x extension, the extra energy appears as oscillations..... Kinetic energy.