Understanding Electric Potential and Field in a Ring of Charge

[WWCY]

Homework Statement

A potentially silly question that I have put off too long to ask, any assistance is greatly appreciated!

The electric field evaluated along the Z axis of a ring of charge centered on the origin and lying on the XY plane is only a function of $z$ and points only along $z$. Since
$$\vec{E} = - \nabla V = - \big( \partial _x V \hat{x} + \partial _y V \hat{y} + \partial _z V \hat{z}\big) $$
Is it right to say that along the $z$ axis,
$$E_z = - \partial _z V$$
and therefore the potential evaluated along the $z$ axis is as such,
$$V = - \int \ dz \ E_z$$

where I can pick the constant to be an arbitrary number?

Thanks in advance!

Homework Equations

The Attempt at a Solution

 

[kuruman]

Sure. Usually the potential is zero at infinity, but it doesn't have to be.

 

[WWCY]

Sure. Usually the potential is zero at infinity, but it doesn't have to be.

Thank you!