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Consider connecting two batteries of different voltages $x$ and $y$ in parallel. Considering a series circuit, the total voltage would be $x+y$. However what happens in this parallel case?

From my understanding, when the two batteries are of the same voltage and connected in parallel, the total voltage remains the same, only the load to each voltage source is halved. Current can only go through one of these voltage sources, so the voltage is the same.

Now when connecting two different batteries in parallel, I have been told that the battery of higher voltage discharges into the battery of lower voltage, until they are both of the same voltage. Essentially the higher voltage battery charges the lower one. I have no idea what this means, or furthermore why it even occurs. Could someone please explain this to me? It seems that connecting different batteries in parallel is a bad idea.

Would connecting different batteries in series also be bad?

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Unequal batteries should never be used in parallel or in series. You can approximate the consequences using Ohm's law, treating the battery as a pure potential difference in series with a resistor of the battery's internal resistance. (Most batteries are at least a little bit rechargeable, since chemistry works both ways, so Ohm's law will only approximate their behavior under external EMF.) In any case, if you have a potential difference across a resistor - which you'll have, either in series or in parallel, if the batteries have different potential difference - you have current through the resistor and therefore power dissipated by the resistor as heat. This can melt or ignite your battery.

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