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  1. Is the voltage of a battery simply the work necessary to move a unit charge from one extreme of the battery to the other?

  2. If this is the case, then when calculating the voltage of a battery the length of such battery should be taken into account, right?

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    $\begingroup$ The voltage of a voltaic cell or battery of such cells is determined by the chemistry in the cell(s). In other words, the specific oxidation and reduction reactions, the electrodes and so on. Length, per se, does not factor in: a 1.5V D cell is longer than a 3V button cell. $\endgroup$
    – Ed V
    Nov 27, 2023 at 20:09

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  1. The voltage of a battery is indeed related to the work necessary to move a unit charge from one terminal of the battery to the other. In more technical terms, it is a measure of the electric potential difference between the two terminals. However, the length of the battery doesn't directly influence this voltage.
  • The work done to move a charge between two points in an electric field is independent of the path taken; it depends only on the properties of the electric field itself. This is why, in the case of a battery, the voltage is determined by the electric field generated by the chemical reactions within the battery, not by the physical length of the battery.
  1. Voltage is an intrinsic property of the electrochemical processes happening inside the battery, determined by the materials and the chemical reactions involved. It's not a function of the physical size or length of the battery. For example, regardless of size, a typical alkaline AA, AAA, C, and D battery will all have a nominal voltage of about 1.5 volts, because they are based on the same chemistry.
  • The size of the battery more directly affects its capacity (measured in ampere-hours, Ah) and the amount of current it can deliver. Larger batteries can generally store more energy and provide higher currents for longer periods of time, but the voltage remains defined by the chemistry of the battery.
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  1. Essentially, yes. The charge needs to be taken from one terminal to the other via a path on which it encounters no resistance and not through the battery itself, which has an internal resistance. Then $$\text{emf}, \mathscr E=\frac{\text{work done on charge}}{\text{charge}}$$ The size of the emf is determined by how many cells constitute the battery and by the chemistry of the cells.

  2. The charges that arise on the terminals of the battery set up a conservative field: The work needed per unit charge to take a 'test' charge from one terminal to the other is independent of the route taken! If we take it by a longer route, the mean electric field component along the route will be lower.

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Is the voltage of a battery simply the work necessary to move a unit charge from one extreme of the battery to the other?

One must be careful when using the term voltage as its use often produces ambiguities.

The emf of a battery is the work done in taking unit positive charge round a complete circuit which includes the battery.
It is a measure of the amount of chemical energy which has been converted to electrical energy within the battery and depends on many factors including the composition of the battery electrodes, the electrolyte between the electrodes, the previous usage of the battery, etc $\dots$

The potential difference across the terminals of the battery is the work done in taking unit positive charge between the terminals of the battery.

So what did you mean by voltage, emf or potential difference?

If you connect a battery to a resistor of resistance $R$, and measure the potential difference across the terminals, $T_1,\,T_2,$ of the battery, $v$, and also the current passing through the circuit, $i$, as the resistance of the resistor is varied and then plot $v$ against $i$ you will get a series of data points as shown below.

enter image description here

You can then model the real battery as an ideal battery with emf $\mathcal E$ and internal resistance $r$ as shown below.

enter image description here

For the straight line portion of the graph the relationship between the potential difference across the terminals of the battery and the current through the battery is $v= \mathcal E -r\,i$.
The model breaks down if the currents are too large or current is drawn from the battery over a prolonged period of time.

You will note that the physical size of the battery has not been mentioned.
Larger batteries of the same type have more chemicals inside them so although the emf is the same they can deliver larger currents for longer periods of time.

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Voltage is W/Q in the external circuit, not between the electrodes. E.M.F is W/Q throughout the circuit including the internal circuit. E.M.F = (R eq. + r)I. As the length between the electrodes affects the internal resistance (r), it is affected by distance between electrodes.

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