Why do batteries in series add voltage and not current ? How does the first battery increase the voltage of the second,i.e. do we model it as the first pushing an electron that actually goes through the second battery appearing at the second's anode ?
Voltage is a potential difference between two points. So, the first battery just creates a higher starting point for the 2nd one. I guess an analogy for this would be two identical step stools placed on top of one another. It is not that the 2nd one is suddenly getting taller, it is just placed on top of the 1st one producing a taller total height.
Why do batteries in series add voltage and not current?
It's the law. Specifically, Kirchhoff's Voltage Law (KVL).
How does the first battery increase the voltage of the second
But, I don't think you're correctly interpreting what it means to "add voltage". One battery does not increase the voltage of the other.
If you place a voltmeter across either series connected cell, you'll still measure the nominal cell voltage.
Only by placing the voltmeter across the series combination will you measure the sum and, again, this must be so by KVL.
An ideal battery is a dc voltage source and an internal resistance in series. This resistance determines the maximum current it can provide. If you add them in series they can deliver the same current as you add the resistors and the sources at the same time. If you put them in parallel you get the same voltage but the resistance gets reduced (Ohm's law), so you get more current.
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