# What does "grounded" conducting sphere placed in an initially uniform electric field mean? Why the resultant field change?

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.

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?

• 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. Commented May 3, 2022 at 14:06