Conductor held at constant potential

Question

What does it mean for a conductor to be held at constant electric potential? So if my reference point is infinity, and I integrate inwards over the electric field to any point on the conductor, I will always get the same $V$?

So does a conductor held at constant potential mean that the electric field outside of it never changes? Does the charge distribution on the conductor have to be constant for it to be held at constant potential?

In the method of images section of the book I'm reading, a charge is placed over a grounded ($V=0$) conducting plane. The charge above the place induces opposite charge on the nearby surface of the conductor. If the conductor is held at constant $V = 0$ (grounded), doesn't that mean there is no net charge on the conductor? If you ground something, charge either flows into the ground or negative charge flows from the ground into the conductor. How can you have induced charge on this grounded conducting plane?

In summary, what has to be constant for a conductor to be held at a constant potential. $V$ can be determined by the line integral of the $\vec{E}$ field or the charge distribution on the conductor. So if something is held at constant $V$, then the device maintaining the constant $V$ ensures either constant $\rho$ or constant $\vec{E}$?

Answer 1

First of all, if in your problem you have to do with conservative fields, the potential on any equipotential surface can be calculated as

$$V(\vec r_2) = \int_{\vec r_1}^{\vec r_2} \vec E d \vec {\ell} \tag{i}$$

where $\vec E$ is the electric field, $d\vec{\ell}$ the element of path length with its respective direction, and you take the potential $V(\vec r_1)$ as zero.

Now, I am taking your questions, one by one.

1) "So does a conductor held at constant potential mean that the electric field outside of it never changes?"

The surface of a conductor doesn't need to be held at a constant potential, it is at a constant potential. But if you connect the conductor to the earth, the convention is that this potential is zero, i.e. there is an international convention that it is taken as reference. About the electric field outside the conductor it may change, leaving though the potential of the conductor unchanged, as long as the result of the integral ${(i)}$ doesn't change.

2) "Does the charge distribution on the conductor have to be constant for it to be held at constant potential?"

I repeat the answer to the previous question, you can make whatever changes you want, as long as the integral doesn't change the potential of the conductor doesn't change. Specifically, the charge on the conductor can change, but other charges in the configuration may also change, so, the answer depends on configuration.

3) " If the conductor is held at constant $V=0$ (grounded), doesn't that mean there is no net charge on the conductor?".

No, it doesn't. Imagine a capacitor with one plate connected to the ground and another plate to the positive pole of a battery whose negative pole is also connected to the ground. The positive charges that gathers on the upper non-grounded plate of the capacitor attracts an equal, but opposite in sign, charge on the lower plate of the capacitor. Where from comes this charge? From the earth.

4) "So if something is held at constant $V$ , then the device maintaining the constant $V$ ensures either constant $\rho$ or constant $\vec E$?"

I believe that this question is elucidated by the former answers.