# Charging of a capacitor by battery

I know that a battery does work when it takes positive charge from one plate of capacitor and deposits it to the other plate, thereby creating equal and oppositely charged plates having electric field in between them.However I am unable to understand how this process works.

Let us consider a battery with EMF E connected to an uncharged capacitor and a resistance R. When the circuit is closed, an initial current equal I=E/R is established in the circuit which drops exponentially. In a small time interval dt, a positive charge dq=I*dt will be deposited on one plate of capacitor, which will attract same amount of negative charge on the other plate and hence establish an electric field between them. The electric field of battery doesn't do any work initially since the capacitor is uncharged in the beginning. I believe that later if battery adds more charge to the already present charge, it will have to apply force against the electric field of already deposited charges and thus do work in the process. Is my assumption correct?

However if I place a mechanical source of energy between plates of an uncharged capacitor such as a conveyor belt (instead of a chemical one which is a battery) which takes a positive charge dq from one plate and deposits it on the other plate, it has to do work against the electric field between positive and negative charge of magnitude dq to create a charge separation.

So how come a battery doesn't do any work to create an initial charge separation of dq while a mechanical source of energy does? Are the two processes of charging capacitors different or am I missing something in the battery scenario?

Let us consider a battery with EMF E connected to an uncharged capacitor and a resistance R. When the circuit is closed, an initial current equal I=E/R is established in the circuit which drops exponentially.

That is correct.

In a small time interval dt, a positive charge dq=I*dt will be deposited on one plate of capacitor, which will attract same amount of negative charge on the other plate and hence establish an electric field between them.

The electric field is not established because the positive charge deposited on one platee "will attract the same amount of negative charge on the other plate". It is established because the positive terminal deposits positive charge one one plate while at the same time the negative terminal removes and equal amount of positive charge from the other plate. (What's really being deposited and removed are electrons, but that's another matter). In effect, the battery does work to separate the charge on the capacitor plates.

The electric field of battery doesn't do any work initially since the capacitor is uncharged in the beginning.

Correct, because the voltage across the uncharged capacitor is zero. The potential difference $$V$$ between two points is defined as the work required per unit charge to move the charge between the two points. Since the initial voltage across the capacitor is zero, no work is initially required to move the charge. As the voltage increases, the work required increases. But the maximum work per unit charge the battery can do is its own emf, which is why charging stops when the capacitor voltage equals the emf of the battery.

I believe that later if battery adds more charge to the already present charge, it will have to apply force against the electric field of already deposited charges and thus do work in the process. Is my assumption correct?

Yes, because as the charge is separated the voltage increases, and thus the work required per unit charge to move more charge increases. In effect, it gets harder to pull positive charge off of a more negatively charged plate the more negative it becomes and to push positive charge onto a more positively charged plate the more positively it becomes, since the attraction and repulsive electrical forces increase.

However if I place a mechanical source of energy between plates of an uncharged capacitor such as a conveyor belt (instead of a chemical one which is a battery) which takes a positive charge dq from one plate and deposits it on the other plate, it has to do work against the electric field between positive and negative charge of magnitude dq to create a charge separation.

It's a bit of an odd mechanical analogy, but OK. @Farcher has given you an excellent fluid mechanics analogy.

So how come a battery doesn't do any work to create an initial charge separation of dq while a mechanical source of energy does? Are the two processes of charging capacitors different or am I missing something in the battery scenario?

I'm not sure why you think the conveyor does initially do work. Any mechanical work associated with moving the mass of the charge off and onto the plates would be negligible compared to the work against the force of the electric field.

In any event, like the battery, the conveyor also does not have to do any work to create the initial charge separation. That's because the attractive and repulsive forces that have to be overcome are initially negligible.

Hope this helps.

• Thankyou for your answer. I now understand that positive charge pushed by positive terminal doesn't attract negative charges but their motion is simultaneous. Therefore when initially the plates are uncharged the work is zero. – user14598090 Dec 13 '20 at 1:49
• I now feel that the conveyor belt example is perhaps not accurate for describing the charge buildup since some work is still done to create an initial charge separation of dq between plates when conveyor works against the electric force between +dq which is moving on conveyor and -dq on the plate, although the amount of work done is small. Am i right? – user14598090 Dec 13 '20 at 1:58
• @user14598090 You are indeed right. The initial work is not exactly zero but for all practical purposes is small enough to say it is zero. That is why the rate of charge movement (current) is a maximum when a switch is first closed on the circuit. – Bob D Dec 13 '20 at 2:15

The battery acts like a charge pump driven by chemical energy.
The charge pump creates a potential difference (pressure difference) across it which can move charges in a connecting circuit.
Omitting talk of electrons moving, the charge pump (battery) causes positive charge to move from one plate of the capacitor (which then has an excess of negative charges) to the other plate (which then has an excess of positive charge) which creates a potential difference (pressure difference) across the plates.
This continues until the pressure difference created by the charge pump is equal to the pressure difference across the capacitor.

So overall chemical energy has been converted to energy stored in the capacitor.

"However if I place a mechanical source of energy between plates of an uncharged capacitor such as a conveyor belt (instead of a chemical one which is a battery) which takes a positive charge dq from one plate and deposits it on the other plate, it has to do work against the electric field between positive and negative charge of magnitude dq to create a charge separation."

Is this still for the initial increment of time after the switch is closed? If so no work is needed, for the same reason that you acknowledged that no work was needed initially to charge the capacitor electrically (only to do work against resistive forces in the resistor). As the capacitor gains charge, work would be needed to give each plate more charge, whether that work is done mechanically or electrically.