The electrons always takes the easiest way in a circuit, right?
So in a parallel circuit, why does the electrons flow through all parts of the circuit and not just the one with the least resistance?
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The ratio between currents flow in paths that are in parallel, is a function of the ratio between the equivalent resistances of the paths. This means that unless one path has zero or infinite amount of resistance, there will be non-zero currents flow into all paths that are in parallel. However, in general, more current will flow into the path with least amount of resistance and you can show this physically.
Imagine a river flowing towards a fork. The water on the left side of the river does find it easier to go on the left fork and the water on the right does find it easier to go on the right.
Now imagine that the right branch looks like more of a side street than a fork. But if there is a slow down in the forward direction and the side branch goes downhill then you would still expect some of the water to take the right turn. Specifically the water on the right finds it easier to go to the right.
The same thing happens with a circuit. Charges on the surface of the wires provide a net field that guides the current in different parts (right or left) to tend towards the particular branches of the parallel circuit.
So each part of the wire has some current and each part does find it easier to go on a particular branch of the circuit.
Ohm's law helps here.
When a current $I$ flows through a resistor $R$, it develops a voltage $V=I\cdot R$. If there is less current, there is less voltage.
When the voltage developed is equal to the voltage applied, you reach equilibrium. So when current can "choose" between a high resistance branch and a low resistance branch, it will divide in such a way that the voltage developed across each branch is the same. This means less current through the high resistance branch, and more current through the low resistance branch - so that the voltage across each branch ends up being the same.