What is the connection between the two definitions of voltage of potential difference?

Question

When we were introduced to the concept of potential difference, we were given the example of water flow between two beakers through a pipe. Water flows through the pipe from a beaker which has higher quantity of water to second beaker which has lower quantity of water.

I understood. There is more water in one beaker so it flows to the other beaker. I then thought since there are more electrons at one terminal of the battery and less electrons at the other terminal, electrons flow. This was easy to understand as potential difference. But then this definition came: "Potential difference is defined as the work done in moving a charge from one point to another point".

I am unable to find how the two concepts are related to each other. And if the second definition is the actual meaning of the term, then I fail to see how it can cause current to flow, since it is the work done by an electron. It does not provide a reason for the flow of charge. Please can someone explain this to me.

Answer 1

When you use the water circuit analogy the volume of water is equivalent to the charge and the difference between the heights of liquid $h$ is equivalent to the potential difference.
A water pump in the circuit is supplied energy and raises the potential (energy) of the water with the equivalent cell using an electrochemical reaction to raise the potential (energy) of mobile charge carriers.

The water pump in raising unit mass of water a vertical height $h$ will do $gh$ amount of work and so the difference in the potential between the water at the top and the bottom is $gh$.
The cell increase the potential of unit positive charge when going from the negative terminal to the positive terminal at the expense of $V$ amount of chemical energy. $V$ is called the potential difference between the two cell terminals.

Your connected beakers example filled with different heights of water, say $h_1$ and $h_2$ with $h_1>h_2$, is equivalent to two capacitors being connected together with the potential difference across the plates of each capacitor being different, say $V_1$ and $V_2$ with $V_1>V_2$.
The final state of the beakers will be the level of water being the same in each beaker because water has flowed from beaker 1 to beaker 2 and the final state for the capacitors being the potential differences across each of the capacitors the same because charge has flowed from capacitor 1 to capacitor 2.

Answer 2

Your confusion stems from a misunderstanding of the fluid analogy. In the fluid analogy, the only work in the system is done by gravity, ad potential is gravitational potential (i.e. height). If you give the water a path to flow through, it will flow to the lowest possible height. Similarly, if you give electrons a conducting path to flow through, they will flow to the lowest possible potential. Potential difference, in this sense, is analogous to difference in height.

In the other sense, potential difference in the fluid analogy is the amount of work done on a unit mass of fluid when it is moved from one point to another. This work is done by or against gravity, and so only depends on the height of the water, so the analogy remains consistent. Similarly, electric potential difference is the amount of work done on a unit charge when it is moved from one location to another.