Electric Fields Around Batteries

Battery / Capacitor transformation

A) A battery will charge a capacitor.

B) If you increase the distance between the capacitor plates, the field will remain.

C) There is a capacitance between the electrodes of a battery.

D) Therefore a battery has an electric field around it.

I just realized that the negative terminal is the entire battery chassis. Therefore the field will look something like this.

Capacitor plates with a higher charge at the plates to make it behave sort of more like 3D, correct me if I’m wrong