It is interesting to look in more detail at what happens in the circuit when an extra battery is added.
First of all, think about the charged particles as they are driven by the push of a single battery through the filament of the bulb. Each charge is both driven and retarded as it moves through the array of fixed ions. The interaction with the ions retards the charge. The driving forces are provided by the remote action of the battery. As the ions are spaced out the charge travels a short distance before interacting with the next ion. Therefore the motion is not steady, as the retarding and driving forces vary with the position of the charge.
With an extra battery, the positive terminal of the battery becomes more positively charged and the negative plate becomes more negatively charged, creating a greater force on the charged particles.
As a result:
- More charged particles pass through the filament each second
- More energy is shifted by each passing charge.