Conservation of Charge: Explaining the Point Rule

[hidayah]

Why does the point rule arise as a consequence of conservation of charge?

 

[Antiphon]

Why does the point rule arise as a consequence of conservation of charge?

What is the point rule?

 

[jtbell]

I was wondering what the "point rule" is, myself. Then I saw hidayah's other question about the (Kirchhoff's) loop rule for circuit analysis. I suspect he's asking about what the books I've used call the "junction rule", i.e. in a multibranch circuit the sum of the currents entering a junction must equal the sum of the currents leaving the junction. This also applies to any point along a branch of a circuit, just think of it as a trivial junction with one current coming in and one going out.

I also suspect that we have a pair of homework questions here.

OK then, conservation of charge says that charge can be neither created nor destroyed. If we had more current (charge) leaving a junction than entering it, where would the extra charge have to come from? Likewise, if we had more charge entering a junction than leaving it, where would have to happen to the extra charge? What does conservation of charge say about the possibilities of these things?

 

[Antiphon]

OK then, conservation of charge says that charge can be neither created nor destroyed. If we had more current (charge) leaving a junction than entering it, where would the extra charge have to come from? Likewise, if we had more charge entering a junction than leaving it, where would have to happen to the extra charge? What does conservation of charge say about the possibilities of these things?

Well, I don't see a problem here. The extra charge would come down
one of the wires. This is what happens when you charge a capacitor,
no?

 

[Zelos]

thats a bad exemple. in the atom world the charge is allways conserved. like if a + charged particle is created a - is created aswell. in all nuclear proceses this happen. like in beta decay, there a electron is created but also a proton. so a neutron that is not charged become a proton and a electron with a total charge of nothing. in beta + decay a positron is created but a proton becomes a neutron and then it still the same amount of charges.

 

[Antiphon]

thats a bad exemple. in the atom world the charge is allways conserved. like if a + charged particle is created a - is created aswell. in all nuclear proceses this happen. like in beta decay, there a electron is created but also a proton. so a neutron that is not charged become a proton and a electron with a total charge of nothing. in beta + decay a positron is created but a proton becomes a neutron and then it still the same amount of charges.

You're missing the point. We're talking about a junction of ideal wires.
The question is why cannot charge accumulate at the junction.
Conservation of charge should have nothing to do with it since I can
have a source of charges (battery) ready to push them into the junction.
There is overall conservation, but it doesn't follow from that that charge
can't accumulate in the junction.

A charged capacitor has more charge than an uncharged one.

 

[pervect]

You're missing the point. We're talking about a junction of ideal wires.
The question is why cannot charge accumulate at the junction.
Conservation of charge should have nothing to do with it since I can
have a source of charges (battery) ready to push them into the junction.
There is overall conservation, but it doesn't follow from that that charge
can't accumulate in the junction.

A charged capacitor has more charge than an uncharged one.

It seems to me the point is that idealized wires dont' have capcaitance. (Nor inductance). So they can't store charge.

The next most accurate approximation is to assign lumped values of capacitance and inductance to wires, which are usually incorporated into the circuit elements. So this reduces to the first case in terms of analysis.

The most accurate approximation is to treat wires as distribuited systems incoroporating capacitance and inductance - with simple geometries, this means transmission lines. One does not use Kirchoff's current law in this case, one uses instead reflection coefficients.

More accuracy than this requires a field approach, where one thinks not in terms of wires and components, but Maxwell's equations.

 

[jtbell]

Well, I don't see a problem here. The extra charge would come down one of the wires.

But then it would be counted as part of the charge entering the junction originally!

One of the unstated assumptions of the junction rule is that charge enters or leaves a junction of wires only via the wires.

 

[rbj]

You're missing the point. We're talking about a junction of ideal wires.
The question is why cannot charge accumulate at the junction.

Oh! we're talking about Kirchoff's Current Law for electrical circuits! i was wondering what this was about when i first read it.

Conservation of charge should have nothing to do with it since I can
have a source of charges (battery) ready to push them into the junction.

i s'pose a little bit of charge (very small) can accumulate at a node (or junction) or anywhere else, but if the amount of accumulated charge gets to be large enough, tremendous forces (remember the Coulbomb force constant is about 10¹⁰ in SI units) will build up and push most of that collected charge to places where it doesn't exist.

KCL and KVL have hydrostatic analogs. because like charges really don't like each other (they're really heterosexual or homophobic and $k = 9 \times 10^9 \frac{Nt \ m^2}{C^2}$), electrical current is sort of incompressible just like some fluid like water is.

 

[Antiphon]

Pervect gets the prize.

The ideal wire has no capacitance nor inductance. It can't store charge and
doesn't produce a magnetic field.

Note that in a junction of real wires, charge can accumulate and this
merely raises the voltage via $$V = Q/C$$.

like charges really don't like each other (their really heterosexual)

Hmm... Since electrons like both protons and positrons, does this mean
they are trisexual?

 

[rbj]

Pervect gets the prize.

fine with me.

The ideal wire has no capacitance nor inductance. It can't store charge and
doesn't produce a magnetic field.

and the reason for that is... ?

Note that in a junction of real wires, charge can accumulate and this
merely raises the voltage via $$V = Q/C$$.

not very much. not even microcoulombs. if the current differential is a microampere, in one second a microcoulomb will build up (and because of conservation of charge a microcoulomb of the opposite polarity will build up somewhere else) and then you get to tell me what the forces on those built up charges will be.

Hmm... Since electrons like both protons and positrons, does this mean
they are trisexual?

no, protons and positrons are the same gender but different species. like kinky sex with aliens, but i think that the Pope and Pat Robertson would still approve as long as the electron and positron got married.

 

[Antiphon]

and the reason for that is... ?

..because if they did, the circuit diagram would show them as inductors and capacitors.

not very much. not even microcoulombs. if the current differential is a microampere, in one second a microcoulomb will build up (and because of conservation of charge a microcoulomb of the opposite polarity will build up somewhere else) and then you get to tell me what the forces on those built up charges will be.

I agree. The typical junction of three wires coming together is probably in the range of a few picofarads relative to another nearby wire.

no, protons and positrons are the same gender but different species. like kinky sex with aliens, but i think that the Pope and Pat Robertson would still approve as long as the electron and positron got married.

Hmm... When an electron and proton marry, they produce a stable hydrogen
atom. But if they are forced together, they could become a single neutron.
Maybe ths is an argument against arranged marriages.

When an electron and positron join up, they soon explode in big flash.

I think electrons are Husbands, Protons are wives, and positrons are
misteresses!

 

[rbj]

..because if they did, the circuit diagram would show them as inductors and capacitors.

and that is a problem for what reason? (let's have a physical answer.)

I think electrons are Husbands, Protons are wives, and positrons are
misteresses!

i think you have that right.


i'm curious, what other tiny particles are charged? i can only think of those three.

 

[Crosson]

Pervect gets the prize.

Here is a more physical answer, which is from Grifith's:

Imagine that charge is accumulating at a junction. Fewer positive charges going out then going in. The electrostatic force of this excess charge is actually self corrective, because it decreases the flow of positive charges into the juntion, and increases the flow of positive charges out of the junction.

 

[rbj]

Imagine that charge is accumulating at a junction. Fewer positive charges going out then going in. The electrostatic force of this excess charge is actually self corrective, because it decreases the flow of positive charges into the juntion, and increases the flow of positive charges out of the junction.

it's what i tried to say in post #9 above.

 

[pervect]

not very much. not even microcoulombs. if the current differential is a microampere, in one second a microcoulomb will build up (and because of conservation of charge a microcoulomb of the opposite polarity will build up somewhere else) and then you get to tell me what the forces on those built up charges will be.

What I was trying to point out was that Kirchoff's current law is really a engineering approximation. It works very well for low frequencies. At radio frequencies, one first starts to run into the problem of unwanted coupling between nearby circuit elements. This could be regarded as the start of the breakdown of the law, though it's more commonly not viewed that way.

At Ghz frequencies though (where cellphones routinely operate nowadays), circuit board layout is very important, and the capacitance and inductance of the circuit board are essential parts of the circuit and important to it's operation. Long traces on the board need to be modeled as transmission lines, or perhaps even more exotic elements such as "directional couplers" (google if you're interested for more details). You don't really design circuits using Kirchoff's current law at these sorts of frequencies - wires are not passive elements. The circuit board layout is part of the circuit.

Addressing your calculation of the force between charges - If you think on the timeframe of not seconds, but nanoseconds (a complete cycle at 1 Ghz), you can see that there is not enough time for much charge buildup to occur.

 

[rbj]

What I was trying to point out was that Kirchoff's current law is really a engineering approximation.

you can say that again.

It works very well for low frequencies. At radio frequencies, one first starts to run into the problem of unwanted coupling between nearby circuit elements. This could be regarded as the start of the breakdown of the law, though it's more commonly not viewed that way.

At Ghz frequencies though (where cellphones routinely operate nowadays), circuit board layout is very important

it's a big deal even for 1/2 GHz or 100 MHz applications (like the computers of 5 years ago). all of those issues you mention (transmission line modeling of PC board traces) apply to computer boards. at least modern ones.

now i got to go and see what Tide is saying (and resist the temptation to go about it another round).