# What means "A grounded conductor has potential equal zero"?

When I have a force field $$\boldsymbol{\vec{F}}$$ it tells me how the momentum of the particle will change given in a position on space.

When we talk about an electric field $$\boldsymbol{\vec{E}} = \boldsymbol{\vec{F}}/q$$ it seems to be a generalization of this concept since now it's independent of the other charge, now I can tell how would, the behavior of any charge under the influence of this electric field, be like. When I talk about an electric potential $$V$$, it seems to be again a generalization of work, since $$\int\boldsymbol{\vec{E}}\cdot dr = \Delta V$$. And now it makes sense why we always talk about the difference of potential because what we can measure in nature is the variation of energy.

When we say that something has an electric potential $$V$$ normally we put the point of reference on infinity to make $$V(x) - V(\infty) = V(x) - 0$$, but we still talk about the difference of potential $$\Delta V$$.

If I'm thinking correctly about it, my question is:

When I have a charge at distance $$d$$ from a conductive grounded plane like the image below, by induction, the plane will have a charge equal to -q.

When I say that the conductor is grounded and has a potential equal to zero, is the same as saying that when a take any other charge at infinity distance of the plane and I bring it to this plane the work done should be zero? this seems to me to be a little counterintuitive

Note that the potential at infinity, which you call $$V(\infty)$$, is only meaningful to talk about if $$\lim_{|\mathbf r|\rightarrow \infty} V(\mathbf r)$$ is well-defined. If you have an infinitely large, uniformly charged plane, for example, then this limit is not well-defined because it $$V(\mathbf r)$$ diverges to infinity if you walk in the direction perpendicular to the plane while remaining constant if you walk in some direction parallel to the plane.
If we postulate some charge $$q$$ at a distance from ground, this does not mean that ground has the opposite charge; that opposite charge could be anywhere.