Metal sphere in an electric field

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SUMMARY

The discussion centers on the behavior of a metal sphere in an external electric field, specifically addressing the concept of symmetry and equipotential surfaces. The sphere, while inherently an equipotential object, does not maintain spherical symmetry due to the influence of the external electric field, resulting in cylindrical symmetry instead. The induced charges on the sphere's northern and southern hemispheres create a constant potential across the XY plane, which can be set to zero by choice. Understanding these principles is crucial for accurately analyzing the electric field interactions with conductive objects.

PREREQUISITES
  • Understanding of electric fields and potentials
  • Familiarity with equipotential surfaces
  • Knowledge of symmetry in electrostatics
  • Basic principles of charge distribution on conductors
NEXT STEPS
  • Study the concept of cylindrical symmetry in electrostatics
  • Learn about induced charges on conductors in external electric fields
  • Explore the mathematical formulation of electric potential and equipotential surfaces
  • Investigate the implications of boundary conditions in electrostatic problems
USEFUL FOR

This discussion is beneficial for physics students, educators, and anyone studying electrostatics, particularly those focusing on the behavior of conductive materials in electric fields.

Clara Chung
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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
 
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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.
 
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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.
 
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