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I have some trouble understanding the equations that you get from Kirchoffs law when there are several batteries in a circuit. Consider a circuit with two batteries like the attached.

Kirchoffs laws just gives ohms law for the two circuits as were they completely independent:

U1 = RI1

U2 = RI2

But how does this make sense. I always understood Kirchoffs laws as a rule developed by the basic idea that current flows such that no charge piles up anywhere. Can anyone explain why the above exactly insures this property? Why can none of the current from one battery flow through the other?

Kirchoffs laws just gives ohms law for the two circuits as were they completely independent:

U1 = RI1

U2 = RI2

But how does this make sense. I always understood Kirchoffs laws as a rule developed by the basic idea that current flows such that no charge piles up anywhere. Can anyone explain why the above exactly insures this property? Why can none of the current from one battery flow through the other?