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Let us say we have the following setting:

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Two conductors with potentials $\phi _{1}$ and $\phi_{2}$ (with the zero set to infinity) enclosed within a hollow conducting shell having zero charge.

How can we practically set the potential of the shell to zero without changing $\phi_{1}$ or $\phi_{2}$ ?

If we could connect the sphere to the negative side of a battery having a potential $\phi '$ where $\phi '$ is the potential present on the sphere before connecting the battery, that should set the sphere's potential to zero, but I think this would also alter $\phi _{1}$ and $\phi_{2}$, wouldn't it?

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In theory, one could fix absolute potential of the big shell to zero by connecting it via conducting wire with a body that has known absolute potential 0. In practice we don't know any such body, there is no measuring equipment that could determine absolute potential. Nobody has ever moved charge from its position $\mathbf r_0$ to infinity to measure absolute potential of position $\mathbf r_0$. It is too difficult to move things to infinity.

Instead, we use some body of assumed constant potential as the reference body of potential 0, instead of infinity. Usually Earth's ground plays this role well because it is conductive and has great capacitance.

So you could set potential of the big shell to 0 by grounding it. This will change potential of the inner bodies, so to get at prescribed values $\phi_1,\phi_2$, some forced transfer of charge from/to the inner bodies will be required.

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