Metal sphere in an electric field

[Clara Chung]

Homework Statement

The question is exactly the same as this question

https://physics.stackexchange.com/questions/250297/metal-sphere-in-a-uniform-electric-field

However, it says "Then by symmetry the entire xy plane is at potential zero" in the book. I don't understand this.

Homework Equations

The Attempt at a Solution

I understand the sphere is an equipotential... but I don't get the idea of how the symmetry of the sphere is related to the region outside the sphere

 

[vela]

Because of the external electric field, you don't have spherical symmetry in this problem anymore. Instead, you have a cylindrical symmetry. You can't just look at the sphere alone to determine what symmetry you have. You have to consider the entire situation.

 

[timetraveller123]

Because the induced charges is similar on the northern and southern hemisphere. If it were just the sphere with same charge distribution without the external electric field the xy plane would be zero potential because the potential would cancel out in pairs. With the addition of the external electric field you just add constant potential to the XY plane. You are free to choose the constant.