# When does the relation $V = Ed$ hold?

In my Physics class, we are studying electrostatic potential, and now that we have reached the concept of capacitance, we are using the relation: $$V = Ed$$

We derived it from the definition of potential for an infinite plane, and I understood that perfectly. However, my question is, how come we are also applying it to cables (cylindrical symmetry) and other geometries?

All help appreciated.

The relationship is that the electric field is (minus) the potential gradient.

If the electric field is constant then the potential gradient is constant and so the potential varies linearly with position.

For a cylindrical symmetric situation the electric field and hence the potential gradient is not constant and so the relationship that you quoted is not valid.

If you have two coaxial cylindrical conductor of radius much, much larger than their separation then a small part of them do look like parallel plates.
So the electric field between those cylinders will be approximately uniform and the potential gradient will be approximately constant.

In general $V=\int \vec E\cdot d\vec\ell$ so this reduces to $Ed$ when the electric field is constant. In particular, this will not hold for cylindrical geometry since the electric field of a cylinder goes like $1/r$ and is thus NOT constant. Likewise, in the spherical case, the field goes like $1/r^2$ and not not constant either. Indeed, the potentials outside a cylinder and outside a sphere typically go like $\log(r)$ and $1/r$, respectively.