In the video above the man put a 200 thousands volt ball near a non-charged one and a very tiny current appear between them. But I think with high electric potential difference between them, the ions should be pushed with higher force since F=q.E and thus more charge particles moving through a region per time which means higher current.
In reality there are many batteries that have the same voltage but different supply different currents. I know it has something to do with power but in my logic the battery is like a pump pushing electrons through conductor and it's voltage determines how much force it apply on the electrons to get them from cathode to anode which means with the same voltage they'll deliver the same amount of electrons. Somebody told me that it's like a narrow and a big water pipe under the same pressure, the bigger will let larger amount of water go through, however, if the pipe get narrower, isn't the current be faster?
Speaking of power. I still don't understand what exactly is it. A battery have a specific power but how does it control the voltage and the current? P = U1 x I1 = U2 x I2 = ... So if we rise the voltage, what causes the current to decrease? Just like the Earth and Mars, putting two objects in the same height on two planets, the object on Earth will have higher PE and thus fall faster just like ions travel at higher speed between higher voltage. So how is it possible that having a high potential difference cause the current to be smaller?