Is this statement of conservation of charge circular?

According to Wikipedia:

A closed system is a physical system that does not allow certain types of transfers (such as transfer of mass and energy transfer) in or out of the system.

According to my textbook, the principle of conservation of charge is:

The algebraic sum of all the electric charges in any closed system is constant.

Isn't this circular logic? In terms of charge, a "closed system" is one in which charge can neither exit nor enter. If the charge neither exits nor enters, then of course the sum thereof stays constant.

Or is the principle saying that the only way for the sum of charge in a system to change is via transfer of charge in or out of the system? (In this case, wouldn't it make more sense to state the principle as "charge can neither be created nor destroyed"?)

• I'd remind everyone that comments are meant for suggesting improvements or requesting clarifications on their parent post. I deleted a few comments that weren't along those lines. – David Z Aug 28 '18 at 0:05
• @MaxvonHippel OK. I proposed an edit to include that in the question, and I've deleted my comment. – David Richerby Aug 28 '18 at 17:46
• Perhaps think about the contrapositive version of the statement: If the sum of charge changes, some charge must have come from somewhere else, or gone somewhere else. Or in other words, you can't pull it out of nowhere. – immibis Aug 29 '18 at 5:59
• Your phrase "then of course ...(etc)" is a non sequitur. It's only valid because it is a property of electric charges, but not a property of all phenomena. For example (1) What becomes of the gravitational fields in a proton-antiproton collision? (2) What about the Entropy of a closed system? – DanielWainfleet Aug 29 '18 at 11:33
• According to Noether's Theorem, charge conservation and gauge symmetry of the em field are indeed "sort of" circular. Gauge symmetry (also) depends on the boundary conditions of the system under consideration. In the global, and in your local case, the same condition applies, i.e. charge has to vanish at the boundary of the system. Hence, the symmetry, hence the charge conservation. – StefG Sep 1 '18 at 7:59

You're right that it's a bit circular as stated. The more rigorous way to state a conservation law is something like:

The rate of change of [quantity] in a bounded system is equal to minus the rate at which [quantity] leaves through boundaries of that system.

A "closed system" is then a system for which both of these rates are zero, i.e., the [quantity] is not moving through the boundary of the system.

The version you propose, "[quantity] can neither be created nor destroyed", is closer to this more rigorous statement. But the rigorous statement is a bit stronger than this. If a charge were to suddenly teleport across the room, without passing through the points in between, this would satisfy your version of the statement; but it would not satisfy the rigorous version of the conservation law above. Moreover, it's perfectly possible in particle physics for charges to be created or destroyed, so long as equal amounts of positive and negative charges are created or destroyed. Your version of the statement would seem to outlaw these events, but the rigorous version does not.

• Any teleportation of charge would violate the "neither create nor destroyed" rule in some reference frames. – Beta Aug 27 '18 at 14:09
• If it's understood as "no net charge can be created or destroyed", then it stands. – Acccumulation Aug 27 '18 at 15:54
• @Acccumulation: if that was a reply to me, then no, it doesn't. Teleport an electron, and in some reference frames an electron ceases to exist, then at a later time an electron appears from nothing; temporary destruction violates the rule. – Beta Aug 28 '18 at 0:22
• @Beta No, it was directed at Michael Seifert., who said "Moreover, it's perfectly possible in particle physics for charges to be created or destroyed, so long as equal amounts of positive and negative charges are created or destroyed. Your version of the statement would seem to outlaw these events, but the rigorous version does not." The rule "no net charge can be created or destroyed" would not outlaw the events in question. – Acccumulation Aug 28 '18 at 20:22
• How is this more rigorous than the textbook version? How is the textbook version circular at all? – aquirdturtle Aug 28 '18 at 20:25

No, the definition is perfectly correct and not circular at all.

For example, consider the principle of "conservation of sound", which states that if no sound enters or exits a closed system, then the total amount of sound in that system is constant. That is false, because I can clap my hands. Sound is not conserved, even if you don't let any come in from outside, because you can make it. You can't do that with charge, so the statement is nontrivial.

• I like your metaphor but I don't know enough physics to really understand it because I don't really understand where charge "comes from". But I think that's another question entirely, probably (and probably one my textbook can answer). Thanks for contributing to my understanding :) – Max von Hippel Aug 27 '18 at 5:20
• -1: this answer misses the question from afar. The circularity is contained in the definition of closed system (according to the question), not in the fact that one quantity is or is not produced. – gented Aug 27 '18 at 12:55
• @gented No it isn’t. There’s nothing circular about defining a closed system to be one where nothing passed through the walls. That’s precisely what I did for sound — it does not automatically follow that sound is conserved. – knzhou Aug 27 '18 at 17:33
• You didn't understand my comment, again. I am not objecting your definition of closed system, I am objecting that your answer doesn't actually answer the question (which isn't about how to define a closed system, rather about how to get rid of the circularity in the definitions that the question mentions). – gented Aug 28 '18 at 9:11
• @gented So we agree that defining a closed system is fine, and we agree that the subsequent definition of charge conservation is not circular, so what’s the problem? What exactly do you think is circular? – knzhou Aug 28 '18 at 17:18

You could imagine a theory of electromagnetism in which the charge of 'fundamental' particles in the theory (let's stick to protons) is allowed to change with time. In these types of theories, closed systems do not have the property that the net charge within them is constant. The principle of conservation of charge is a statement that says that no such theories can describe reality as we observe it.

Your last paragraph is correct in that the principle of conservation of charge does imply that the only way for the sum of charge in a system to change is via transfer of charge in or out of the system. This does not preclude processes in which, for example, we may have a neutral particle decay into a positively charged particle and a negatively charged particle (in practice, beta decay), so the statement "charge can neither be created nor destroyed" depends a bit on what exactly you mean by 'created' and 'destroyed'.

These statements are not circular but equivalent, you can assume one is true and the other follows. That is:

if

A: A closed system is a physical system that does not allow charge transfer.

and

B: The algebraic sum of all the electric charges in any closed system is constant.

are true then we can conclude that:

C: Net electrical charge can neither be created nor destroyed.

Similarly, if A and C are true, we can conclude B.

This is similar to geometry, where you can either postulate the existence of parallel lines and then prove that the sum of angles in a triangle is 180°, or you can postulate that the sum of the angles is the same for every triangle and prove that parallel lines exist.

Circular reasoning is a fallacy which appears when someone is trying to prove a statement using an equivalent statement as assumption. This is not the case for the laws of conservation, which are postulated, not proven.

• Should C be Net electrical charge ...? Per what other answers have described RE annihilation of charge? Thanks. – Max von Hippel Aug 29 '18 at 21:03

Yes, it is saying that the only way to change the sum of charge in a system is via transfer into or out of the system. "Charge can neither be created nor destroyed" is not strictly accurate, though. For instance, if a positively charged positron and a negatively charged electron collide, they can "annihilate," leaving behind uncharged photons. This doesn't change the net charge, but it does "destroy charges."