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What happens to the electronic flow inside two conductors when they are connected to the terminals of a battery but not mutually? Why do they reach at the same potential as that of the terminals?

When a conducting plate (say of a capacitor) is connected to a wire which is connected to a battery terminal, the charge appears only on the plate but not on the whole system (wire plus plate). Why?

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The electric potential is given by the integration of the eletric field over a path. If you think in a battery (a car battery as example) there is a field leaving the cathode and entering the anion which lose its strength for greater distances. When you connect conducting wires to both terminals, they will (almost intantenously) ionize the air around them in a way that it cancels the electric field inside. From this moment on, any integral of the eletric field that you take from any point of one wire to any point of other will lead you to the same value or, in other words, the same potential diference. Thanks to this, we say that the wires and the terminals are in the same potential.

Concerning the second part, the charge will appears all over the system indeed, we say them show off just in the plates for the convenience of the model (the ideal case). As the plates are very close to each other the field between is pretty high and as consequence more charges will gather there, making the rest of the circuit despicable.

Here is a good reference about the air ionization: https://www.youtube.com/watch?v=DOMs7mYm_zs&ab_channel=ElectroBOOM

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When a conducting wire is connected to the terminals of the battery, a potential difference is created between the ends of the conductor. This potential difference setup and electric field throughout the conductor

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