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 is a property of the configuration and depends only on the positions of the charges, not on the path taken to assemble them (i.e., 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; the field does positive work.

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

• System of Multiple Point Charges: The total potential energy is the sum over all unique pairs.

Example: Three Point Charges in a Right Triangle

For three charges arranged as a right triangle, the total potential energy is the sum of the energies for each pair:

Changing the arrangement changes the distances and thus the total energy.

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 , its potential energy changes by .

Example: Kinetic Energy Change

If a charge of +2e moves from a point at 500.0 kV to 200.0 kV:

Example: Electron Accelerated by Potential Difference

• Given m/s,

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:

Uniform Electric Field

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

Potential Due to a Distribution of Point Charges

Superposition Principle

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

Example: Find the Potential at (0, a/2)

Given three charges at specified positions, calculate the distance from each to the point (0, a/2) and sum their contributions using the above formula.

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.

• The electric field is always perpendicular to equipotential surfaces.

• Equipotential surfaces are useful for visualizing electric fields and potentials.

Gauss’ Law and Applications

Electric Flux

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

• Formula:

• For a closed surface:

Gauss’ Law

Gauss’ Law relates the electric flux through a closed surface to the charge enclosed:

• Useful for calculating E when symmetry is present (spherical, cylindrical, planar).

Example: Point Charge

• For a point charge at the center of a sphere:

Example: Infinite Line of Charge

• For a long charged rod, use a cylindrical Gaussian surface: where λ is charge per unit length.

Properties of Conductors

Charged Conductors

• All excess charge resides on the surface.

• Inside the conductor, the net charge density is zero.

• The electric field inside a conductor is zero.

• Just outside the surface, E is perpendicular to the surface.

• The surface of a conductor is an equipotential.

Example: Conducting Spherical Shell

• If a charge +q is placed at the center, the induced charge per unit area on the inner and outer surfaces can be calculated using Gauss’ Law.

Summary Table: Key Equations and Concepts

Additional info:

• These notes expand on the provided slides and images, adding definitions, formulas, and examples for clarity and completeness.

• Some equations and explanations are inferred from standard physics curriculum for introductory electricity and magnetism.