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

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Yes, if the sizes are different.

Capacitance of capacitor $\mathrm{C}=\frac{\mathrm{Q}}{\mathrm{V}}$, where $\mathrm{Q}$ is the charge of conductor and $\mathrm{V}$ is the electric potential of conductors.

For given charge potential $\mathrm{V} \propto \frac{1}{\mathrm{C}}$. So two adjacent conductors carrying the same charge of different dimensions may have different potentials.

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  • [AIEEE 2012]

No current flows between two charged bodies connected together when they have the same