Electric Potential and Gauss’ Law

Overview

This section covers the fundamental concepts of electric potential energy, electric potential, equipotential surfaces, and Gauss’ Law. These topics are central to understanding electrostatics in college-level physics.

Electric Potential Energy

Definition and Properties

• Electric potential energy () is the energy stored in a system of charges due to their positions relative to each other.

• This energy depends only on the configuration of the charges, not on the path taken to assemble them (the force is conservative).

• By convention, at infinite separation () for two point charges.

Formula for Two Point Charges

• For two point charges and separated by distance :

• Where is Coulomb’s constant.

Example: Proton and Electron

• For a proton and electron brought closer together:

• Decreasing (bringing charges closer) decreases potential energy; the field does positive work.

Like Charges

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

Multiple Point Charges

• For a system of point charges:

• Sum over all unique pairs of charges.

Example: Three Charges in a Right Triangle

• Potential energy is the sum of the energies for each pair:

Electric Potential

Definition and Units

• Electric potential () is the electric potential energy per unit charge:

• Measured in volts (V), where .

• In atomic physics, the electron-volt (eV) is commonly used: .

Potential Due to a Point Charge

• For a collection of point charges:

Potential Difference and Energy Change

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

Example: Kinetic Energy Change

• If a charge moves from kV to kV:

Example: Electron Accelerated by Potential Difference

• Electron accelerated from rest to m/s:

• Note: The electron moves from low to high .

Finding Electric Potential from Electric Field

General Relationship

• If the electric field () is known, the potential difference between points and is:

• This integral is path-independent for electrostatic fields.

Uniform Electric Field

• For a uniform field, the potential changes linearly with distance.

Potential Due to a Distribution of Point Charges

• For point charges at various positions:

• Example: Find the potential at for three charges arranged on the and axes.

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 and points in the direction of greatest decrease in potential.

Gauss’ Law

Statement and Application

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

• Useful for calculating electric fields when there is symmetry (spherical, cylindrical, planar).

• Choose a Gaussian surface so that and are parallel and is constant over the surface.

Example: Point Charge

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

Example: Infinite Line of Charge

• For a long, straight line of charge with linear charge density :

Properties of Conductors

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

• The electric field inside a conductor is zero ().

• The surface of a conductor is an equipotential.

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

Summary Table: Key Equations and Concepts

Additional info:

• For continuous charge distributions, integrals replace sums in the formulas for and .

• Equipotential surfaces are useful for visualizing electric fields and understanding the motion of charges in electrostatic situations.