Question

What distance do you use in the W=Fs formula? Is it the distance the applier of force moved or the object who feels the force? If it's case

1) That would mean a stationary proton moving another electron would be doing no work, but that's false. If it's case
2) Then when climbing up stairs, I would be moving relative to the stairs and the stairs would be applying a normal force hence doing work. But that's false stationary stairs do no work.
It is never the source (the applier) that we consider. We only consider the properties of the object that is being influenced. In all laws and relationships (just think Newton's laws of motion) it is always only the object that is being influenced that we consider. So suggestion 1) is incorrect.
The distance r you should in W=F.r is the distance that the influenced object moves while the force F is acting!.
In your staircase example, you are exerting a force on the stairs with your legs, and notice that the normal force produced by the stairs does not cause the particles of your foot to move. What actually moves is your upper body while your thigh muscles are extending/compression. Only when you have taking the next step is the first foot lifted, caused by a muscle force once again and not by the normal force from the stair.
In general human motion does not give good, clear examples because there are so many factors and so many moving parts involved in the human body machine. Rather, a good, simple example might simply be pushing a cart - maybe a gold miner's rail cart on the tracks. In such example you have a clear force applied and a clear displacement happening.
In the equation W= →F⃗→s⃗ the term s refers to the displacement of the object on which the force is applied.
Then when climbing up stairs, I would be moving relative to the stairs and the stairs would be applying a normal force hence doing work. But that's false stationary stairs do no work.
When you climb stairs, it's your own muscular effort that counteracts gravity, effectively performing work. The stairs themselves, although providing necessary support, do not contribute to this work. They apply a normal force directly opposing your weight, but crucially, this force acts at a right angle to your upward trajectory. Given that this force does not align with your direction of movement, it results in no work being done by the staircase.
For the calculation of work done by a Constant Force, we generally use W=F⃗ .S⃗
, Where S⃗ being the The amount of displacement of the object while the force is/was being applied, because as soon as you remove the force the object will stop to accelerate anymore and will move with a constant velocity, in order to perform work on something we require some interaction between the subject and object and force is the way to do that in classical mechanics because that's how the mechanical energy will get transferred from one object to another object. So if there is no force and just displacement then it won't count as work done.

Now talking about the example you took for a stationary proton (considering the case of an atomic nucleus) and an electron, the proton is what we call 'Subject' here and there is no such rule that the stationary subject can never apply force and hence cannot do work without moving, i.e the motion any object (in this case it is electron) only depends on the direction and amount of force being applied on itself (object), we don't have to care about what subject is doing, though the subject actions may or may not impact the direction or magnitude of force being applied (for example, the coulomb's law. The force being applied by any charge on other charge depends on the distance between them.)

For the example of stairs you took, for that I would say No the stairs aren't doing any work in that case, the normal force being applied by the stairs is just helping us to stop our leg at that point, i.e the real work is done by us, our muscles, by pushing us upwards.

Answer 1

The distance used in the work equation W=Fs is the displacement of the object on which the force is applied. The work done by stairs or a stationary proton is dependent on the displacement and force applied to the object, not on the movement of the force source itself.

Step-by-step explanation:

In the context of physics, when we talk about work W=Fs (Work = Force x Distance), the distance s refers to the displacement of the object on which the force is applied, not the distance that the source of the force has traveled. Therefore, in the example of pushing a cart, where you exert a force on the cart and it moves along the tracks, the work calculated would be based on the distance the cart moves, not how much the person pushing it has moved.

Considering work done during stair climbing, it is the climber who does the work against gravity. The stairs, while providing a reaction force (normal force), do not move or displace, hence they do no work. For work to be done, there must be displacement in the direction of the component of the applied force.

For a stationary proton affecting an electron, the work done can still exist if the electron moves due to the force exerted by the proton. When climbing stairs, the work is done by your muscles as they exert a force that moves you up against gravity. The normal force from the stairs is perpendicular to your movement, so it does not contribute to the work done on you by the stairs.