The drift velocity does not depend on the length or the cross sectional area of the wire, when dealing with a macroscopic (ordinary, everyday life) wire. However, if the wire is, say, too short, e.g. comparable to the average distance a charge carrier travels before undergoing a collision, then it might begin to depend on the wire length, but for all practical intents and purposes a wire won't be that short.
The reason v does not depend on the wire cross sectional area is that the ratio I/A is constant (assuming the applied electric field within the wire is not changing), also called the current density, denoted by J=I/A. So, for example, if A doubles, I will also double (wire capacity doubles), keeping J constant.
Because of electromagnetic forces, all of the electrons in the wire are displaced towards A with a certain velocity causing a positive current towards B.
The electrons have a small drift velocity, not moving much.
Although your light turns on very quickly when you flip the switch, and you find it impossible to flip off the light and get in bed before the room goes dark, the actual drift velocity of electrons through copper wires is very slow. It is the change or "signal" which propagates along wires at essentially the speed of light.
A single electron does not go from A to B. Think of it as each electron pushing the next one, and the signal travels with the velocity of light ,maximum, down the wire.
This drift of electrons heats up the highly resistive filament wire in the bulb and makes it glow.
True.
But I don't understand why they are able to move back and forth. Specially if length of the wire was large, say 3 * 10^8 meters, then would the movement of electrons on one end of the wire be "in sync" with movement of electrons on the other end?
Why not? When the field changes at A and B the change propagates by electrons moving back and forth over an average position. Similar to water waves, the atoms do not move much from their position, the energy is transferred atom by atom. In the case of electric fields and electrons the field is built up at the microscopic scale by the motion in situ of the electrons, in a sinusoidal way.
Very long wires enter the realm of special relativity and the limit of the velocity of light in transferring effects of fields.
Best Answer
A larger diameter wire has lower resistance.
A lower resistance wire produces less heat for a given current flowing through it.
The less heat the wire produces, the lower its temperature will be while operating.
Therefore a larger-diameter wire will be able to carry a larger current before it heats up enough to melt or scorch its insulation or ignite other objects near it.
Anything that lowers the resistance of the wire will improve its current carrying capability (aka ampacity). So using a material with lower resistivity will also help.
However it might be cheaper to use a slightly higher resistivity material like aluminum than a lower resistivity material like copper, but simply make the aluminum wire thicker. Unfortunately, aluminum wire has other issues (work hardening and oxidation) that caused problems when we tried to use it for residential wiring in place of copper.