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Why does current follow the path of least resistance? Will all current pass through a wire with $0$ resistance in a junction leaving other resistive wires with no current?

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Why does current follow the path of least resistance?

Current doesn't follow the path of least resistance. Please forget that you ever heard that and if you hear someone else say that, respond with something like this:

If there are two parallel connected resistors with resistance $R_1$ and $R_2$ respectively, the current divides between the two paths such that the path with smallest resistance has largest current through.

This is simply due to Ohm's law and, in particular, the fact that parallel connected circuit elements have identical voltage across (by definition!).

With voltage $V$ across, the current through each resistor is given by Ohm's law:

$$I_1 = \frac{V}{R_1}$$

$$I_2 = \frac{V}{R_2}$$

and now it's clear that the smaller resistance has the largest current through. Denoting the total current $I = I_1 + I_2$, we easily derive the current division formula:

$$I_1 = I\frac{R_2}{R_1 + R_2}$$

$$I_2 = I\frac{R_1}{R_1 + R_2}$$

which again shows that the smaller resistance has the largest current through. Note that if (just) one of the resistances is zero, then all of the current is through the zero ohm resistor. For example, if $R_1 = 0$ then

$$I_1 = I\frac{R_2}{0 + R_2} = I$$

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Yes. Think of it like a pipe system with water.

If the pipe splits in two, one having a "squeezed" constriction and the other being much wider, then most water will go through the wide one. Infinite resistance corresponds to fully closing the constriction (no water passes), and 0 resistance corresponds to having the wide pipe infinitely wider than the other (all water would go there).

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