Why does current chose the easy way? Is there a proper mathematical formula about why current tends to go over a resistance which is smaller?
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1$\begingroup$ We generally don't do homework here. So start from the beginning, why would current go through the most resistance? $\endgroup$– zeta-bandCommented Mar 18, 2019 at 20:15
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$\begingroup$ First draw a circuit with options. $\endgroup$– my2ctsCommented Mar 18, 2019 at 20:25
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1$\begingroup$ Possible duplicate of Electricity takes the path of least resistance? $\endgroup$– BioPhysicistCommented Mar 19, 2019 at 1:48
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2 Answers
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Current doesn't always go through a resistance which is lower. If you have two ohmic resistors in parallel, both of them will have current through them. Yes, there will be a larger current through the more conductive resistor (lower resistance), but there is current in the less conductive resistor (more resistance), too. So you can't say that current always goes through the lower resistance.
Ohm's Law, which is typically used to describe this situation does not tell us why this happens. It only tells us that it will. It is a phenomenological/experimental relationship. It merely tells us a relationship between voltage and current through an ohmic circuit element.
To answer why one would have to understand some condensed matter physics.
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$\begingroup$ You can use KVL to prove that two resistors in parallel must both see the same voltage... $\endgroup$ Commented Mar 18, 2019 at 21:45
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From Ohm's law $$ j = \sigma E $$ you directly see that to say "current always chose to go to lower resistance" is not accurate. $j$, which is more precise to describe currents, exists as long as there is an electric field around.
So it's better to say currents tend to go where the resistance is lower. In fact, for a parallel circuit, we have a distribution of current $$ i_j = \frac{G_j}{\sum\limits_i G_i}i_{total} $$ so the current of jth branch is proportional to its $R^{-1}$, which confirms the trend of currents we were considering about.
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$\begingroup$ What about when Ohm's law is not valid? $\endgroup$ Commented Mar 19, 2019 at 4:59