Electric Fields

Introduction to Electric Fields

Electric fields are fundamental to understanding the forces that charged particles exert on each other. This chapter explores the definition, calculation, and visualization of electric fields, as well as their behavior in different physical situations.

Coulomb's Law and Forces Between Charges

Coulomb's Law

• Coulomb's Law quantifies the electrostatic force between two point charges:

where , and are the charges, and is the distance between them.

• The force is attractive if the charges are of opposite sign and repulsive if they are of the same sign.

Example: Three Charges in Two Dimensions

• To find the net force on a charge due to multiple other charges, calculate the force from each charge using Coulomb's Law and add the forces as vectors.

• Forces in two dimensions require resolving each force into components and then summing the components:

• Example Result: at from the x-axis.

The Electric Field

Definition of Electric Field

• The electric field at a point is the force per unit charge experienced by a small positive test charge placed at that point:

• SI Units: newton per coulomb (N/C)

• A positive test charge is used by convention, so the direction of the electric field is the direction of the force on a positive charge.

Calculating Electric Field from Point Charges

• The electric field due to a point charge at a distance is:

• The field points away from positive charges and toward negative charges.

• The electric field does not depend on the test charge used to measure it.

Superposition Principle

• Electric fields from multiple sources add as vectors:

Example: Electric Field of a Point Charge

• Given , , :

Example: Electric Fields May Cancel

• For two positive charges and separated by distance , the point where their fields cancel can be found by setting their field magnitudes equal and solving for the position:

• Solving yields possible positions where the net field is zero.

Electric Field Lines

Properties of Electric Field Lines

• Electric field lines (or lines of force) provide a visual map of the electric field in space.

• Field lines point away from positive charges and toward negative charges.

• The density of lines indicates the strength of the field: more lines per area means a stronger field.

• Field lines never cross and always begin on positive charges and end on negative charges.

• The number of lines leaving or entering a charge is proportional to the magnitude of the charge.

Examples of Field Line Patterns

• Single point charge: radial lines outward (positive) or inward (negative).

• Electric dipole: lines emerge from the positive charge and curve toward the negative charge.

• Parallel plate capacitor: uniform field lines between plates, except near the edges.

Drawing Electric Field Lines: Common Mistakes

• Field lines should not begin or end in empty space.

• Lines should not cross each other.

• The number of lines should reflect the relative magnitudes of the charges.

The Electric Field Inside a Conductor: Shielding

Electrostatic Equilibrium in Conductors

• At equilibrium, any excess charge resides on the surface of a conductor.

• The electric field inside a conductor is zero under electrostatic conditions.

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

• Conductors shield their interiors from external electric fields (Faraday cage effect).

Example: Induced Charges in a Hollow Conductor

• If a charge is placed at the center of a hollow, neutral conductor:

  • A charge of is induced on the interior surface.

  • A charge of appears on the exterior surface.

The Parallel Plate Capacitor

Uniform Electric Field Between Plates

• The electric field between two large, parallel plates is uniform (except near the edges):

where is the surface charge density, is the plate area, and is the permittivity of free space.

Summary Table: Key Electric Field Equations

Additional info:

• Examples and conceptual questions throughout the notes reinforce the application of electric field concepts to real and idealized physical systems.

• Understanding the vector nature of electric fields and forces is crucial for solving multi-charge problems.