# Battery in non-closed circuit with charged body and Earth

Looking at the clips on how the battery works, I understand that eventually one of the ends releases the electrons which then move through the conductor to the other end, to continue the reaction. Now I am struggling with the following thought experiment, if negative battery end connected to earth and positive connected to negatively charged body with electrons surplus (created by means of friction for example), will this non-closed circuit produce the current flow?

Earth <--- [-][battery][+] <--- [ e- charged body e- ]

• It is a closed circuit if you account for capacitance, which you must do to calculate the flow. Commented Jul 9 at 10:22

With no current passing through it a battery maintains a constant potential difference across its terminals via an electrochemical retraction.

In the situation you have described a current will flow until there is no potential difference between the negative terminal of the battery and the Earth and also there is no potential difference between the positive terminal of the battery and the charged body.
In other word the steady state with no current flowing will be achieved when the potential difference between the terminals of the battery is the same as the potential difference between the Earth and the body.

In your example, taking the Earth as the zero of potential, if the potential of the body was greater than that of the the potential of the positive terminal of the battery, electrons would flow from the Earth to the body until the steady state of equality of potential was reached.

• I am kinda trying to think in terms of chemical reaction. For it to happen the positive battery side must be provided with electrons (that what charged body is for) on other hand, the earth removes the surplus of electrons from negative side. In this case why wouldn't flow continue until charged body is depleted from electrons? And given the charge is big enough, why wouldn't the flow continue for quite time limited only by chemical reaction rate ? Commented Jul 9 at 8:39
• There will be a transfer of charges between the Earth and the body until the steady state at which time no further transfer of charges occurs. If you have a battery which is not connected to anything the electrochemical reaction within the battery moves charges until there is a potential difference between the terminals, the emf of the battery. So you have a terminal with a net negative charge and a terminal with a net positive charge. $\dots$ Commented Jul 9 at 9:00
• $\dots$ Those net charges on the terminals set up an electric field which opposes the action of the electrochemical reaction so in the steady state is no more charges move between the terminals. Commented Jul 9 at 9:01

Assume that the negatively charged body is a conductor, so that the negative charge $$Q$$ can move freely from and to it. Then the potential difference of the body with respect to earth $$V$$ is negative and determined by its capacitance $$C$$: $$V=Q/C$$ When you connect the positive electrode of the battery to the body, the negative charge (electrons) of the body will flow through the battery to earth until the body ends up with a positive charge (missing electrons) $$Q$$ determined by the capacitance $$C$$ and the positive battery voltage $$V_B$$ $$Q=C V_B$$

• So there will be a flow untill charged body depleted of electrons, speed of which determined by reaction speed? Commented Jul 9 at 8:40
• @Boris You are right! The speed of the electron charge depletion is determined by the time constant $\tau =RC$ of the capacitance $C$ and the internal resistance $R$ of the battery. Commented Jul 10 at 9:33
• @Boris For any macroscopic body, the chemical reaction speed will be much much faster than the discharge speed determined by the RC time constant, which governs the discharge process speed. Commented Jul 10 at 10:08

In practice, there will be almost no motion of the electrons. A transient electromagnetic wave will propagate outside the conductors and the battery at something like the speed of light. The electrons in the metal and the ions in the battery will very slightly adjust their positions to support this. A tiny amount of chemistry will accompany this.

How tiny? Simple objects on a human scale have capacitances of picofarads relative to their surroundings ("ground"): a human body has about $$100\ pF$$. So, assume a $$100\ pF$$ body with a $$kV$$ of electrostatic potential. That's $$(100\ pF)\times(1\ kV)=(100\ nC)$$ of charge.

Meanwhile, the chemical capacity of a one amp-hour battery to pump charge is $$3.6\ kC$$. So, the fraction of the battery capacity represented by this is $$(100\ nC)/(3.6\ kC)=10^{-10}$$. One ten billionth.

• Thanks a lot, clear, but the question was purely theoretical, not assuming the capacitance of the charge body. Assume, it has exactly the number of electrons needed to get 3.6kC of charge to equal the chemical capacity of the battery you have mentioned. Will this not-closed circuit produce the constant current flow? What confuses me is the "not-close d" part which contradicts the basic rule of having current only in closed circuits. Commented Jul 9 at 13:46
• @Boris Without capacitance, the body cannot be charged. This is a physics site: we cannot address self-contradictory hypotheses. Commented Jul 9 at 14:47