Watts - I think I'm getting free energy from a circuit?
Congratulations for having the wit to know something was wrong!







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Parallel and series arrangements of batteries will have the same VAh rating.


  I will effectively have a battery of 20 V with a capacity of 2 Ah.


That's the error. 


In parallel they can supply 1 A each for one hour. This is very easy to visualise if you run one for an hour and then replace with the other.
In series they both run simultaneously and the same current runs through both of them. They will both discharge at the same rate.


This is one good reason to use Wh (watt-hours) for battery comparison. It makes it much easier to compare battery energy storage when the batteries have different voltages. 



In your example you have \$ Capacity = V \cdot Ah = 10 \cdot 1 = 10~Wh \$. With two batteries we have \$ 20~Wh \$ capacity whether in series or parallel.

Our load power is given by \$ P = \frac {V^2}{R} \$.


In parallel \$ P = \frac {10^2}{100} = 1~W \$ as you calculated. \$ Runtime = \frac {Wh}{W} =  \frac {20}{1} = 20~h \$.
In series  \$ P = \frac {20^2}{100} = 4~W \$ as you calculated. \$ Runtime = \frac {Wh}{W} =  \frac {20}{4} = 5~h \$.










simulate this circuit

Figure 2. An alternative view may help. (a) Is your original series circuit. (b) is a direct equivalent. Because the mid-point of the 50 Ω resistors is at the same potential as the mid-point of the battery stack it makes no difference if we connect them together as shown in (c). Now we have two 1 Ah circuits each feeding into a 50 Ω load.