4.3 Equipotential surfaces and their properties

Electric potential is key to understanding energy in electric fields. Equipotential surfaces are where potential is constant, meaning no work is needed to move charges along them. These surfaces are always perpendicular to electric field lines.

Conductors in equilibrium have special properties. Their surfaces become equipotential, with excess charge on the outside. Inside, the electric field is zero. This concept is crucial for understanding how charges behave in conductors.

Equipotential Surfaces

Definition and Properties

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• An equipotential surface represents a set of points in space that have the same electric potential
  • All points on the surface have the same potential energy per unit charge
  • No work is required to move a test charge along an equipotential surface (frictionless movement)
• Equipotential surfaces are always perpendicular to the electric field lines at every point
  • The electric field vector is always directed along the steepest slope of the potential (gradient)
  • The direction of the electric field is from high potential to low potential (opposite to the gradient)
• In a uniform electric field, equipotential surfaces are equally spaced parallel planes (infinite charged sheets)
  • The potential difference between adjacent surfaces is constant
  • The electric field strength is inversely proportional to the spacing between the surfaces

Work and Energy Considerations

• No work is done when moving a charge along an equipotential surface
  • The potential energy remains constant along the surface
  • The kinetic energy of the charge does not change
• Work is required to move a charge from one equipotential surface to another
  • The work done is equal to the potential difference between the surfaces multiplied by the charge ($W = qΔV$)
  • Positive work is done when moving a positive charge from low to high potential (against the electric field)
  • Negative work is done when moving a positive charge from high to low potential (with the electric field)

Equipotential Lines and Diagrams

• In two dimensions, equipotential surfaces are represented by equipotential lines
  • Equipotential lines are curves along which the electric potential is constant
  • Equipotential lines are always perpendicular to the electric field lines
• Equipotential diagrams provide a visual representation of the electric potential distribution
  • Closely spaced equipotential lines indicate a strong electric field (rapid change in potential)
  • Widely spaced equipotential lines indicate a weak electric field (slow change in potential)
  • Equipotential lines can never cross each other (potential is single-valued at each point)

Conductors and Equilibrium

Electrostatic Equilibrium in Conductors

• In electrostatic equilibrium, the electric field inside a conductor is zero
  • Charges redistribute themselves on the surface until the electric field inside becomes zero
  • Any excess charge resides on the surface of the conductor
• The electric potential is constant throughout the volume of a conductor in equilibrium
  • All points inside and on the surface of the conductor are at the same potential
  • The surface of a conductor in equilibrium is an equipotential surface
• The electric field just outside the surface of a charged conductor is perpendicular to the surface
  • The field lines terminate perpendicularly on the surface
  • The magnitude of the electric field is proportional to the surface charge density

Conductors as Equipotential Surfaces

• The surface of a conductor in electrostatic equilibrium is an equipotential surface
  • All points on the surface have the same electric potential
  • No work is required to move a charge along the surface of the conductor
• Equipotential surfaces are always perpendicular to the electric field lines
  • The electric field lines just outside the conductor are perpendicular to the surface
  • The electric field inside the conductor is zero (no field lines inside)
• The shape of a conductor determines the shape of the equipotential surface
  • A spherical conductor produces a spherical equipotential surface
  • A cylindrical conductor produces a cylindrical equipotential surface
  • Irregularly shaped conductors produce equipotential surfaces that conform to their shape