Q.If a conductor has a potential $V \neq 0$ and there are no charges anywhere else outside, then

A non-zero potential $V \neq 0$ with no charges anywhere outside the conductor can only be caused by charge on the conductor itself — that is the only possible source, since the potential of any charge distribution is $V=0$ at infinity, and here there is nothing else around to raise it.

In electrostatic equilibrium the electric field inside the conducting material is $\vec{E} = 0$ everywhere. Apply Gauss's law to any closed surface drawn just inside the conductor's outer boundary:

Since this holds for every such interior surface, no charge can sit inside the body — all of it resides on the outer surface. …

Why the conductor must be charged. Electric potential is measured relative to infinity, where the potential of any finite charge distribution is zero. The problem states there are no charges anywhere outside the conductor, yet its potential is $V \neq 0$. The only thing left that could produce this non-zero potential is charge residing on the conductor itself. So the conductor is necessarily charged — an uncharged, isolated conductor with nothing else around it would sit at $V=0$.

Where that charge can live. In electrostatic equilibrium, the electric field inside the material of a conductor is zero:

This is a basic property of a conductor in equilibrium — any internal field would drive free charges to move until it vanishes. Now draw a closed Gaussian surface entirely within the conducting material, just beneath the outer surface. Since $\vec{E}=0$ everywhere on and inside this surface, the flux through it is zero, and by Gauss's law:

This holds for every such surface, however close to the true outer surface it is drawn. So no net charge can exist anywhere in the interior of the conductor — the entire excess charge is pushed out to the outer surface.

Evaluating each option.

• (a) "There must be charges on the surface or inside itself." True — the charge is on the surface, which satisfies this (deliberately loose) statement. …

Method: Locating an Isolated Charged Conductor's Charge from Its Potential

Use this method whenever a conductor is stated to have a nonzero potential with no other charges anywhere nearby, and you must decide where on/in the conductor its own charge can live.

Steps

Step 1: Identify the only possible source of the nonzero potential

Potential is measured relative to zero at infinity, and any finite charge distribution produces $V=0$ far from itself. If the problem states there are no charges anywhere outside the conductor, yet $V\neq 0$, the only thing left that could cause this is charge sitting on the conductor itself — so the conductor must be charged.

Step 2: Apply the equilibrium result $\vec E_{\text{inside}}=0$

As with any conductor in electrostatic equilibrium, the field inside the conducting material is zero — otherwise free charges there would still be accelerating.

Step 3: Use Gauss's law to rule out interior charge

Draw a closed surface anywhere strictly inside the conductor, just beneath its outer boundary. Since $\vec E=0$ on and inside this surface,

$\oint \vec E \cdot d\vec A = \frac{q_{\text{enc}}}{\varepsilon_0} = 0 \implies q_{\text{enc}}=0.$ …