Electric Fields: Point Charges, Dipoles, Continuous Distributions, and Field Lines

Electric Fields and Their Sources

Introduction to Electric Fields

The electric field is a fundamental concept in physics, describing the influence that electric charges exert on their surroundings. It is a vector field, meaning it has both magnitude and direction, and is measured in Newtons per Coulomb (N/C).

• Definition: The electric field at a point is defined as the force per unit charge at that point: $\vec{E} = \frac{\vec{F}_{\text{elec}}}{q}$

• Units: Newtons per Coulomb (N/C)

• Principle of Superposition: The net electric field from multiple sources is the vector sum of the fields from each source: $\vec{E}_{\text{ext}} = \sum_i \vec{E}_i$

Key Electric Field Models

Electric Field of a Point Charge

A point charge creates a radially symmetric electric field. The field strength decreases with the square of the distance from the charge.

• Formula: $E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2}$

• Direction: Away from positive charges, toward negative charges

• Example: The field around a proton or electron

Electric Field of a Line of Charge

An infinite line of charge produces a field that decreases with distance from the line.

• Linear charge density: $\lambda = \frac{Q}{L}$ (charge per unit length)

• Formula for infinite line: $E = \frac{1}{2\pi\epsilon_0} \frac{|\lambda|}{r}$

• Finite line: $E_{\text{rod}} = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{\sqrt{r^2 + (L/2)^2}}$

• Direction: Perpendicular to the line, away from positive charge

Electric Field of a Plane of Charge

An infinite plane of charge creates a uniform electric field near its surface.

• Surface charge density: $\eta = \frac{Q}{A}$ (charge per unit area)

• Formula: $E = \frac{\eta}{2\epsilon_0}$ (near the plane)

• Direction: Perpendicular to the plane

Electric Field of a Sphere of Charge

A spherically symmetric charge distribution (solid or shell) produces a field outside the sphere identical to that of a point charge at its center.

• Formula (outside): $E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{r^2}$ for $r > R$

• Direction: Radially outward (for positive charge)

• Example: The field outside a charged metal sphere

Electric Dipoles

Definition and Properties

An electric dipole consists of two equal and opposite charges separated by a small distance. The dipole moment $\vec{p}$ is a vector pointing from the negative to the positive charge.

• Dipole moment: $\vec{p} = q\vec{s}$

• SI units: Coulomb-meter (C·m)

• Example: Water molecule is a permanent dipole

Electric Field of a Dipole

• On the axis: $E_{\text{dipole, axis}} = \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3}$

• On the bisecting plane: $E_{\text{dipole, bisect}} = -\frac{1}{4\pi\epsilon_0} \frac{p}{r^3}$

• Direction: Depends on position relative to the dipole

Electric Field Lines

Properties and Interpretation

Electric field lines are a visual tool to represent the direction and strength of electric fields.

• Field lines are continuous curves tangent to the field vectors

• Field lines start on positive charges and end on negative charges

• Closely spaced lines indicate stronger field strength

• Field lines never cross

Continuous Charge Distributions

Linear and Surface Charge Density

For objects with charge distributed over a length or area, we use charge density to describe the distribution.

• Linear charge density: $\lambda = \frac{Q}{L}$ (C/m)

• Surface charge density: $\eta = \frac{Q}{A}$ (C/m2)

• Example: Charge on a rod or a metal plate

Electric Field of a Ring of Charge

• For a ring of radius $R$ and total charge $Q$, the field on the axis is: $E_{\text{ring}} = \frac{1}{4\pi\epsilon_0} \frac{Qz}{(z^2 + R^2)^{3/2}}$

• Field is maximum at a certain distance from the center

Problem-Solving Strategies

Electric Field of Multiple Point Charges

To find the electric field at a point due to several point charges:

• Establish a coordinate system and locate all charges

• Identify the point of interest

• Draw the field vectors for each charge at that point

• Use symmetry to simplify calculations

• Calculate each field using $E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2}$

• Sum the vector components to find the net field

Conceptual Questions and Examples

Direction and Magnitude of Electric Fields

• Field direction is away from positive and toward negative charges

• At points equidistant from two opposite charges, the field may cancel or add depending on geometry

• Acceleration of charged particles in a field depends on their charge and mass

Sample Calculations

• For a point charge: $E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2}$

• For a line of charge: $E = \frac{1}{2\pi\epsilon_0} \frac{|\lambda|}{r}$

• For a ring of charge: $E_{\text{ring}} = \frac{1}{4\pi\epsilon_0} \frac{Qz}{(z^2 + R^2)^{3/2}}$

Summary Table: Key Electric Field Formulas

Source	Formula	Notes
Point charge	$E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2}$	Radial symmetry
Infinite line of charge	$E = \frac{1}{2\pi\epsilon_0} \frac{|\lambda|}{r}$	Perpendicular to line
Infinite plane of charge	$E = \frac{\eta}{2\epsilon_0}$	Uniform field near plane
Sphere (outside)	$E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{r^2}$	Same as point charge
Dipole (axis)	$E = \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3}$	Far from dipole
Ring (axis)	$E = \frac{1}{4\pi\epsilon_0} \frac{Qz}{(z^2 + R^2)^{3/2}}$	On axis of ring

Additional info:

• These notes expand on the brief points and diagrams in the original slides, providing full definitions, formulas, and context for each type of electric field.

• Examples and applications are included for each formula, as well as a summary table for quick reference.