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In Purcell's Electricity and Magnetism, p.116 section 3.3, the author spoke about Laplace's equation and said that the boundary conditions for the potential$\,\phi$ on the surface of the conductor may be fixed:

In a real system the potentials may be fixed by permanent connections to batteries or other constant-potential "power supplies."

Does it mean then if I had a conductor like this one:

enter image description here

I would be able to set the potential $\phi$ on its surface to a given value by connecting it in the way below?

enter image description here

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Only if the battery behaves like an ideal voltage source, which means zero internal resistance, or at least internal resistance much less than any load resistance connected to it.

A real battery has internal resistance not shown in your diagram. If connected to a load in addition to the conductor the potential on the conductor will drop. The key term in the statement is “constant-potential” power supply.

Hope this helps.

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