Electric Potential Energy and Electric Potential

Electric Potential Energy

Electric potential energy (Ue) is the energy stored in a system due to the arrangement of electric charges. This energy depends only on the positions of the charges, not on the path taken to assemble them, because the electric force is conservative.

• Reference Point: The potential energy is often set to zero at infinity for two point charges.

• Formula for Two Point Charges: where k is Coulomb's constant, q1 and q2 are the charges, and r is the separation.

• Change in Potential Energy: Bringing opposite charges closer together decreases potential energy (analogous to a mass falling in a gravitational field).

• Same Sign Charges: If both charges are positive or both are negative, and work must be done by an external agent to bring them closer.

Potential Energy of Multiple Point Charges

For a system of n point charges, the total electric potential energy is the sum over all unique pairs:

• General Formula:

• Example: For three charges arranged in a right triangle, sum the potential energies for each pair using their respective distances.

Electric Potential

Definition and Units

Electric potential (V) is the electric potential energy per unit charge. It is a scalar quantity measured in volts (V), where 1 V = 1 J/C.

• Formula:

• For a Point Charge:

• Electron-Volt: A convenient energy unit in atomic physics:

Potential Difference and Energy Change

• When a charge q moves through a potential difference ΔV, its potential energy changes by .

• Potential Due to Multiple Point Charges:

Examples

• Kinetic Energy Change: If a charge moves between two points of different potential, the change in kinetic energy is:

• Electron Acceleration: For an electron accelerated from rest:

Finding Electric Potential from Electric Field

General Relationship

If the electric field (E) is known, the potential difference between two points a and b is:

• Formula:

Uniform Electric Field

• For a uniform field along the y-axis: Note: The potential is zero when y = 0.

Potential Due to a Distribution of Point Charges

Superposition Principle

The total potential at a point is the sum of the potentials due to each charge:

• Formula:

• Example: For charges at specific coordinates, calculate the distance from each charge to the point of interest and sum their contributions.

Equipotential Surfaces

Definition and Properties

Equipotential surfaces are surfaces where the electric potential is constant. No work is required to move a charge along an equipotential surface.

• Relation to Electric Field: The electric field is always perpendicular to equipotential surfaces and points in the direction of greatest potential decrease.

• Examples:

  • For a point charge, equipotentials are concentric spheres.

  • For a uniform field, equipotentials are parallel planes.

  • For a dipole, equipotentials are more complex and depend on the dipole orientation.

Calculating Electric Field from Electric Potential

Gradient Relationship

The electric field is the negative gradient of the electric potential:

• Formula: In Cartesian coordinates:

• Application: If the potential is known as a function of position, the electric field can be found by taking the spatial derivatives.

Gauss’ Law

Electric Flux

Electric flux (ΦE) through a surface quantifies the number of electric field lines passing through that surface.

• Formula:

• Closed Surface: For a closed surface, the flux is proportional to the enclosed charge.

Gauss’ Law Statement

• General Form:

• Applications: Gauss’ Law is most useful for systems with high symmetry (spherical, cylindrical, planar).

• Example:

  • For a point charge, use a spherical Gaussian surface.

  • For a long charged rod, use a cylindrical Gaussian surface.

Properties of Conductors

Key Properties

• All excess charge resides on the surface of a conductor.

• The electric field inside a conductor is zero when charges are at rest.

• The surface of a conductor is an equipotential.

• The electric field just outside a conductor is perpendicular to the surface.

Applications

• Excess charge placed on a conductor moves entirely to the surface, minimizing potential energy.

• For a conducting sphere or shell, the field inside is zero and the potential is constant throughout the interior.

Summary Table: Key Equations and Concepts

Additional info:

• Examples and diagrams illustrate how to apply formulas to real physical situations, such as calculating the potential energy of three charges or the change in kinetic energy when a charge moves between two potentials.

• Understanding equipotential surfaces and their relation to electric field lines is crucial for visualizing electric fields and potentials.

• Gauss’ Law is a powerful tool for calculating electric fields in symmetric charge distributions, but less useful for irregular geometries.