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I am having trouble making sense of kirchhoffs conservation of current. Like most of the explanation of this concept seems to be that whatever charge/electron come in multiple branches adds up and comes out. However this explanation seems to treat current like its a particle.

However current is rate of electron move across conductor per second, isn’t it? Kirchoff’s conservation of current deals with current not with total number of electrons going in conserved. I know it is conserved just like how you cannot create or destroy matter. I know that ultimately number of electron going in has to equal electron os electron coming ou. But what I am confused about is how can the rate of electrons coming out after two branches meet be the same as current that split when electron was first entering the split branches. Because if one of the resistor in the brancher is larger than other, isn’t there gonna be a decrease in speed (aka current) of how much electrons come out of that particular branch with the higher resistor, so at out of one branch with lower resistor current is coming out faster rate from the branch while in the branch with higher resistor, current is coming out slow? This would then imply that at point when two branches meet again, there is less electron coming out due to the speed lag in one of the branches.

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If you're at a junction where one wire carrying current branches in two, and 10 electrons go into the junction in one second, 10 electrons better come out in that same second, otherwise you have electrons building up in the junction. There's no reason for this to happen, especially since electrons don't like to be near each other.

It doesn't matter how you split the electrons leaving the junction, as long as all 10 come out. You could have 5 go through each branch, or in your example where one branch has a strong resistor you might have 9 go through one and just 1 through the other.

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Conceptually, I used to find thinking of current like water flow in a pipe helpful. The water pressure is the Voltage, the rate of flow is the current and the narrowing of the pipe is the resistors. The amount of water flowing in to any pipe or group of pipes has to equal the amount of water flowing out. Easy to visualize and not likely to ever let you down. Kirchhoff likely used the same imagery to work out his math.

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  • $\begingroup$ I think that Kirchhoff would probably have described it as $\nabla\cdot \vec{J}-\partial_t\,\rho=0$, but this is the same as water in pipes, i.e. what goes in either stays in or comes out again. $\endgroup$ Mar 8 '17 at 0:22

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