Electric Potential & Equipotential Surfaces

Introduction

This study guide covers the concepts of electric potential and equipotential surfaces as presented in a college-level Physics II course. These topics are fundamental to understanding how electric fields influence the energy and movement of charges in electrostatics.

Electric Potential

Definition and Formulas

Electric potential (V) at a point in space is the electric potential energy per unit charge at that point. It is a scalar quantity and is measured in volts (V).

• For a single point charge:

• For multiple point charges:

• For a continuous charge distribution:

• Key Terms:

  • q: Source charge

  • r: Distance from the source charge to the point of interest

  • \epsilon_0: Permittivity of free space

Example: The electric potential 0.1 m away from a C point charge is calculated using the first formula above.

Electric Potential for a Pair of Charges

Potential Along the x-axis

When considering two point charges, the total electric potential at a point is the algebraic sum of the potentials due to each charge.

• For a positive charge (+e) at position : The potential is positive and decreases with distance from the charge.

• For a negative charge (–e) at position : The potential is negative and increases (becomes less negative) with distance from the charge.

• Graphical Representation: The potential as a function of position shows characteristic shapes:

  • For a positive charge: V is highest near the charge and decreases with distance.

  • For a negative charge: V is lowest near the charge and increases with distance.

Example: The potential along the x-axis for a pair of charges can be visualized by plotting V versus x, showing the superposition of potentials from each charge.

Equipotential Surfaces

Definition and Properties

Equipotential surfaces are surfaces on which every point has the same electric potential. No work is required to move a charge along an equipotential surface.

• Equipotential surfaces are always perpendicular to electric field lines.

• Moving a charge along an equipotential surface does not change its electric potential energy.

• Examples include concentric spheres around a point charge or planes perpendicular to a uniform electric field.

Analogy: Equipotential lines on a diagram are similar to contour lines (lines of constant elevation) on a topographic map.

Potential Difference and Electric Field

Relationship Between Potential Difference and Electric Field

The potential difference (ΔV) between two points in an electric field is related to the work done by the field in moving a charge between those points.

• Definition:

• For a uniform electric field: (where d is the displacement parallel to the field)

• The direction of decreasing potential is the direction of the electric field.

Example: In a uniform field of 7.0 N/C, the potential difference over a 1.9 m path at 45° to the field is .

Energy Considerations

Electric Potential Energy and Conservation of Energy

The change in electric potential energy (ΔU) of a charge q moving through a potential difference ΔV is:

• For a positive charge, moving from higher to lower potential decreases its potential energy.

• For a negative charge, moving from lower to higher potential decreases its potential energy.

• Conservation of energy applies: the sum of kinetic and potential energy remains constant if only conservative forces act.

Example: An electron (q = –1.6 × 10–19 C) accelerated through a potential difference of 5000 V gains kinetic energy equal to .

Summary Table: Key Electrostatic Relationships

Key Takeaways

• Electric potential is a scalar quantity related to the energy per unit charge.

• Equipotential surfaces help visualize regions of constant potential and are always perpendicular to electric field lines.

• Potential difference drives the movement of charges and is directly related to the work done by or against the electric field.

• Energy conservation principles apply to charges moving in electric fields.