Electric current is the rate of flow of electric charges across any cross-sectional area of a conductor. The direction of electric current is taken as the direction of flow of positive ions or opposite to the direction of flow of free electrons. Your assumption is not necessary here... Electrons always flow from negative terminal to positive terminal.
$$i=\frac{dq}{dt}$$
When current flows through an electrolytic solution or during the process of electrolysis, The plate towards which positive ions (cations) flow is called the cathode and the plate towards which negative ions (anions) flow is called the anode.
Wikipedia says clearly,
In an electrochemical cell, The electrode at which electrons leave the cell and oxidation occurs is called anode and the electrode at which electrons enter the cell and reduction occurs is called cathode. Each electrode may become either the anode or the cathode depending on the direction of current through the cell. A bipolar electrode is an electrode that functions as the anode of one cell and the cathode of another cell.
So, the convention is totally based on our definition of the direction of current flow that it always flows opposite to the direction of electrons (i.e) electrons can be called as cations or anions depending on the usage. And based on this, we dump our thought that cathode should always be negative, etc...
You've asked some really good questions here. Before starting, I want to first mention that the traditional picture of particles moving through a wire in electostatics is missing some physics; for instance, it ignores the quantum mechanical nature of electrons. The reason we still teach this model is because it captures the main effects (the phenomenon of current) without dealing with microscopic details, but I wanted to warn you that some of the answers will involve physics that is probably not contained in your readings in electrostatics.
To put things in perspective, we now know Newtonian physics is "wrong" (or perhaps more accurately, incomplete), and doesn't give the right answers if, for instance, an object is very small or moving very fast. But we still teach Newtonian physics because it's "good enough" for describing macroscopic objects like cars and baseballs.
Now, to answer your questions,
When electrons start moving through the wire to the positive terminal, do they
all move at once? Because otherwise, while they are moving, they will still
exert repulsive forces on each other? Does this repulsive force affect their
movement?
The microscopic picture of a metal is (crudely) a collection of negative charges, aka electrons, moving through a lattice of positive ions. Indeed, there will be an attraction between these ions and the electrons, and repulsion between any two electrons. Surprisingly, there is also an attractive force between the electrons. The origin of this attractive force is that the electrons attract positive charges around them, and can in some cases lead to the formation of a bound state called a Cooper pair, which are relevant for explaining the phenomenon of super-conductivity, a phase of metals where the resistance is exactly zero. Note, this requires quantum mechanics to do properly, and is extremely subtle.
Shouldn't some of the electrons stay in the wire itself? If, at some point of the
wire, there is not enough repulsive force present, will they stop at all, or
will they reach the positive terminal?
Again, we need a more refined model, in this case statistical mechanics. Before connecting the terminals, the electrons all have a random distribution of energy which manifests itself as temperature. The presence of an electrostatic field causes a net flow of charge, but at the micro level, electrons are colliding and moving in a variety of directions. Often times you will see electrostatics books speak of drift velocity of the electrons, which is a statistical representation of the net flow. A single electron is probably moving much faster than the drift velocity, even perhaps in the opposite direction of the current flow, due to the random thermal energy and the collisions between particles.
Will the shape effect the movement of current? Does it have any effect on the electric field?
In electrostatics, no, but in reality, yes. In mechanics, one has statics and dynamics. In electromagnetism, one has electrostatics and electrodynamics. If you keep learning about electromagnetism, you will soon encounter another field, the magnetic field, and you will learn that the electric fields and magnetic fields are intertwined in such a way that lead you to reconsider the two fields as components of a single entity (hence, "electromagnetism"). In particular, you will learn that current carrying wires produce magnetic fields (Ampère's Law) and that changing magnetic fields can produce EMFs (Faraday's Law). This is a legitimate concern for building real world circuits, and the quantity associated with this effect is called impedance. Impedance is measured in Ohms, like resistance, and depends on the geometry of the circuit.
Will the length of the wire effect the speed of the flow of charges? If we have an infinite length of wire, will charges flow at all?
You're definitely on to something here. The resistance of the wire is proportional to the length of the wire. By Ohm's Law, the current is inversely proportional. The current is proportional to the drift velocity, so the current is inversely proportional to the length of the wire. See http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html#c1 for a derivation.
Best Answer
It makes no difference at all where the variable resistor is connected.
The current will be given by Ohms law. viz
i = V/R.
Regardless of where the resistor is located in the circuit shown the electrons pass through it in the same direction, and the same voltage drop occurs in it.
There are circuits where the order of the components matters but that is due either to a branch in the path making conditions different at different locations in the circuit, or due to special interactions outside the scope of Ohm's law (eg magnetic fields may cause interactions between components. )
. _______________________
Water analogy
Consider this water analogy.
Remember "ALL models are wrong. Some models are useful" - George Box.
A pump with a known pressure/flow rate output characteristic is run at a fixed power input level. A standard table says that it operated with open output into a tank with zero head it will pump 1000 gallons per minutes.
The same table says that if it pumps into a 4" pipe 80 feet long, followed by a 2" pipe 20 feet long followed by a 1" pipe 5 feet long then exits with a 3 foot head above the inlet level it will have say a 100 gallons per minute flow rate.
Note, I made the pipe lengths and diameters up - but each mayt have ABOUT the same hydrilic drop - maybe not, not important.
Now
80' x 4" + 20' x 2" + 5' x 1" + 3 foot head = 100 gpm
What flow rate would you expect if the pipes were kept the same but reordered eg
3' head exit in all cases plus 80 20 5 = 100 gpm
80 5 20 = ?
20 5 80 = ?
20 80 5 = ?
5 80 20 = ?
5 20 80 = ?
I strongly suspect that you'll find that the results will be very close to the same in all cases. In the resistive circuit case the results are exactly the same for ideal resistors and real resistors are close to ideal. In pipes with water here may be some second order effects at boundaries but the example is close enough.
Using Ohms's law:
Voltage drop = current x resistance it flows in.
V = I x R Rearranging
I = V/R
R = V/I
In this case total resistance = R_Resistor + Rwire.
R wire is so low (usually) that it can be ignored BUT even if it cannot be the result is still the same whatever the order.
R_Resistor only - ignore R_wire.
I = V/T = V/ T_Resistor - current will be such that the drop around the circuit balances Vbattery. If the resistive drop is lower than Vbattery then current will increase until it balances. The battery does not "know or care" where the resistor is in the circuit - it outputs (conventional current from V+ and accepts the return at V-.
R_Resistor (Rr) +R_wire Rw
2a Flow is through Rr and then through Rw.
Vrr = IR = I x Rr
Vrw = IR = I x Rw
Vtotal = Vbat = I x Rr + I x Rw = I x (Rr + Rw) ....(1)
2b Flow is through Rw and then through Rr.
Vrw = IR = I x Rw
Vrr= IR = I x Rr
Vtotal = Vbat = I x Rw + I x Rwr= I x (Rw + Rr) ....(2)
BUT Rtotal in circuit = Rr + Rw = Rw + Rr = Rtotal
If
Ra = RA and
Rw = RW and
Vbattery1 = Vbattery2
then the current is IDENTICAL in both cases. then
Equation (1) and (2) are identical. The total drop is the result of the common current through Rr + Rw or Rw + Rr. The battery does not 'know' or 'care' which comes first.
Current which flows depends only on the total opposition to current flow.