# [Physics] Many cells in parallel

batterieselectric-circuitshomework-and-exercises

If two or more cells of unequal voltages are connected in parallel (with the same terminal on the same side), is there a formula that gives the net potential difference?

Also, by Kirchoff's loop law, if we go round in the loop (the internal one, not the external one), we gain $V_1$ going in the direction of current and lose $V_2$ going against it. This implies $$V_1-V_2=0$$ but $V_1\neq V_2$. What does this mean?

Once you have a series resistor in your model, it is easy to see how to compute the voltage obtained when the cells are in parallel: you assume an output voltage $V$ and calculate the net current flowing into our out of each of the cells.
One complicating factor is that, depending on the battery chemistry, the "internal resistance" may be different for the cell when it is charging vs when it is discharging. But with all these simplifying assumptions in place, we would calculate the net (no-load) voltage of cells with no-load voltage $V_i$ and internal resistance $R_i$ with:
$$\sum{\frac{V-V_i}{R_i}}=0\\ V = \frac{\sum\frac{V_i}{R_i}}{\sum\frac{1}{R_i}}$$
As a simple sanity check of this equation, if you had two cells with an internal impedance of 1 Ohm each, the expression for their voltage would be $V = \frac12 (V_1+V_2)$ which intuitively makes sense.