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Consider this example of an electrostatic boundary value problem. Let an uncharged hollow conducting sphere of radius $R$ be placed in an initially uniform electric field, $\vec E=E_0\hat{z}$, along the $z$-axis. Such a uniform electric field is easy to set up by using two large oppositely charged plates kept parallel to the XY plane and separated by a small distance. The electric field will induce charges on the conducting sphere which in turn distorts the resultant field in a region near the conducting sphere. The problem is to find out the resultant potential. Mathematically, I can solve this problem blindly but I don't have a clear idea about how to arrange the specified boundary conditions.

In particular, the problem says, if the sphere is "grounded", the potential on its surface is zero. Of course, being the surface of the conductor, the surface of the sphere will remain at a constant potential. But how does one arrange the set-up so that the potential on the surface becomes zero instead of any arbitrary unknown constant? Feel free to sketch a diagram if it can be explained better.

Addition after DKNguyen's answer

After the discussion with DKNguyen, I'm convinced that after grounding, the net charge on the hollow sphere, given by $Q=\int \sigma dS$, must be zero. But this does not mean that the induced surface charge density should be zero. Does it mean that the distortion in the field near the sphere arise due to the nonzero dipole moment?

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  • $\begingroup$ Remember that all electric potentials are with reference to some standard. When there is a ground, that's the reference. When there is no ground, e.g. the "floating output" of a transformer, the potentials have to be referenced to one or the other output ports of the transformer. $\endgroup$ May 3, 2022 at 14:06

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Find an an infinite charge sink, define that potential as your ground/zero potential, then connect your object to it so it shares the same potential.

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  • $\begingroup$ Sorry if this sounds stupid. Can I define the floor of the laboratory as an "infinite charge sink" and connect the sphere with a metal wire to the floor? Is this what you suggest? Will this bring the sphere and the floor at a common potential? $\endgroup$ May 3, 2022 at 14:13
  • $\begingroup$ @Solidification If your floor was conductive and enormous enough. But the Earth is better (it's called ground for a reason), and the power grid and building grounds have metallic earthing network to better distribute charge to the somewhat conductive soil. Unless you're on a ship, then maybe the ship or the saltwater ocean. $\endgroup$
    – DKNguyen
    May 3, 2022 at 14:15
  • $\begingroup$ Many thanks! In the situation described above, there will be some induced charges on the sphere, and they remain stuck on the sphere even after "grounding" has been done. But if we connect two objects at different potentials (here the sphere and the earth) shouldn't the free charges flow before the potential is equalized? $\endgroup$ May 3, 2022 at 14:23
  • $\begingroup$ @Solidification Equalizing is the whole point. And what happens when you try and equalize with something infinite? $\endgroup$
    – DKNguyen
    May 3, 2022 at 14:24
  • $\begingroup$ Well, I would think that charges will flow to the infinite sink making the sphere neutral. $\endgroup$ May 3, 2022 at 14:32

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