Multi-Battery Circuits

With one battery, current has an obvious direction: out of the positive terminal, through the resistors, back in. Add a second battery and that's no longer guaranteed. If the batteries oppose each other, or sit in different branches of a network, the stronger one can force current backward through the weaker one, pushing charge into its positive terminal instead of out of it. When that happens the weaker battery isn't driving the circuit anymore — it's being driven, absorbing energy instead of supplying it. That's charging, the same thing that happens inside a battery on a charger.

To solve these circuits you stop guessing directions and let the algebra decide. Assign a current variable and an arbitrary direction to each branch. Apply the junction rule (current in equals current out) at every node except one, since the last is redundant. Apply the loop rule (sum of EMFs equals sum of IR drops, with consistent sign conventions for direction of travel) around enough independent loops to get as many equations as unknowns. Solve the system. Any current that comes out negative just means the real current runs opposite to your guess — the magnitude is still correct.

This is the natural extension of single-loop circuit analysis into networks, and it sets up the energy bookkeeping that follows: once you know every current, you can find the power each element delivers or absorbs, and check that total power supplied equals total power dissipated plus power going into charging batteries. It's the same Kirchhoff machinery used throughout circuits, just scaled up to more equations.