Suppose I have a battery of $V$ volts and a circuit is made using it. This battery produces a constant electric field which moves the electrons thus establishing current. Now what will happen to the electric field if I increase the length of wire in the circuit (i.e. make it a longer circuit). Will electric field change? If yes then why, since electric field is produced by the battery having some potential difference, it is only related to the battery i.e. $\Delta V = -\int \vec E.d\vec r $ or $E = -\frac{\Delta V}{\Delta r} $ , here $\Delta r$ is the distance between higher and lower potential point of the battety.
If no then why so ??
Best Answer
Yes
Because you kept the potential the difference the same but increase the distance over which it is dropped.
$\Delta r$ is the distance along the path of integration, which in this case is along the path of the wire (because it's along the path of the wire that you know you have a uniform electric field). If you increase the length of the wire then you must also change the path of integration to follow the new wire, to know that you are following a path with uniform field and with the field vector aligned to the path elements.