The difference in voltage between the ends of the resistor produces an electric field.
Any electrons which are able to move are then dragged towards the positive (+ve) end
and pushed away from the negative (-ve) end. In a vacuum they'd get faster and faster until they smashed into the positive end. However, inside a material they find that the atoms keep getting in the way.
Each time the electrons bump into the atoms of material they tend to lose kinetic energy and bounce off at a random angle. As a result they keep being accellerated by the electric field and 'losing their way' each time they hit an atom. The overall effect is to make them tend to diffuse through the resistor. They squeeze their way passed the atoms, a bit like
toothpaste in a tube when you squash it at one end. The harder you squeeze (apply a bigger electric field) the higher their average velocity. Since the current is just a measure of how quickly charge moves along the piece of material we find that the current we get is proportional to the applied voltage.
If we wanted to, we could formulate a version of 'Ohm's Law' for toothpaste... But then toothpaste isn't used much in electronics!...
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