Q.Can there be a potential difference between two adjacent conductors carrying the same charge?

Concept: Electric Potential — potential depends on both charge and geometry (capacitance). Two conductors with identical charge can still be at different potentials if their shapes, sizes, or surroundings differ.

Reasoning:

1. The potential of a conductor is given by $V = \frac{Q}{C}$, where $C$ is its capacitance.
2. Capacitance depends on the conductor’s geometry (size, shape) and its environment (nearby conductors). …

Why This Isn’t Obvious

A common first instinct is: “Same charge means same potential.” That would be true if both conductors were identical in every way — same size, same shape, same environment. But potential is not a function of charge alone. It’s a function of charge and capacitance: $V = Q/C$. And capacitance depends entirely on geometry.

Think of it this way: a small metal sphere and a large metal sphere, both carrying $1\,\mu\text{C}$ of charge. The small sphere has a smaller capacitance, so its potential is higher. The large sphere has a larger capacitance, so its potential is lower. Connect them with a wire, and charge flows until potentials equalise — proving they were not equal before.

So the question is really: can two conductors with the same charge have different capacitances? Almost always, yes.

Step-by-Step Reasoning

1. Recall the definition of potential for an isolated conductor

   For an isolated conductor, the potential $V$ (relative to infinity) is given by $V = \frac{Q}{C}$, where $C$ is its self-capacitance. Self-capacitance depends only on the conductor’s size and shape — not on its charge. For a sphere of radius $R$, $C = 4\pi\varepsilon_0 R$. For a more complex shape, $C$ is some other constant.

2. Same charge, different geometry → different potential

   If two adjacent conductors have the same charge $Q$ but different capacitances $C_1$ and $C_2$, then:

Unless $C_1 = C_2$, the potentials differ. So a potential difference $V_1 - V_2$ exists.

3. “Adjacent” doesn’t change the physics

   The word “adjacent” might suggest they influence each other via electrostatic induction. That’s true — but it only reinforces the point. When two conductors are near each other, their capacitances are modified by mutual influence (the system has a capacitance matrix). Even if they carry the same net charge, the potential of each depends on both its own charge and the charge of the neighbour. The result is almost always a potential difference.

4. A concrete example

   Take two concentric spherical shells — inner radius $a$, outer radius $b$, with $b > a$. Suppose both carry the same charge $+Q$.

   • The inner shell’s potential (relative to infinity) is $V_{\text{inner}} = \frac{1}{4\pi\varepsilon_0}\left(\frac{Q}{a} + \frac{Q}{b}\right)$.
   • The outer shell’s potential is $V_{\text{outer}} = \frac{1}{4\pi\varepsilon_0}\left(\frac{Q}{b} + \frac{Q}{b}\right) = \frac{1}{4\pi\varepsilon_0} \cdot \frac{2Q}{b}$. …

Method: Comparing the Potentials of Charged Conductors

This method applies whenever you're asked whether two (or more) charged conductors are at the same potential, given information about their charge — not their capacitance directly.

Steps

Step 1: Write the potential of each conductor in terms of charge and capacitance

For an isolated conductor carrying charge $Q$, its potential relative to infinity is

where $C$ is its capacitance — a purely geometric property (size, shape, and the presence of nearby conductors), never dependent on how much charge it actually holds.

Step 2: Identify what is being held fixed and what varies …