Chapter 21: Electric Charge and Electric Field

Learning Outcomes

This chapter introduces the fundamental concepts of electric charge and electric fields, which are foundational to understanding electrostatics and electromagnetism.

• Distinguish between electric force and electric field.

• Visualize and interpret electric fields using electric field lines.

• Calculate properties of electric charge distributions, including dipoles.

Relationship to Maxwell’s Equations

Maxwell’s Equations Overview

Maxwell’s Equations are the fundamental laws governing all classical electromagnetic phenomena. In this chapter, the focus is on static electric charges and electric fields (electrostatics).

• Electric field (\(\vec{E}\)): A vector field representing the force per unit charge at each point in space.

• Charge density (\(\rho\)): The amount of electric charge per unit volume.

• Vacuum permittivity (\(\varepsilon_0\)): A physical constant describing the ability of the vacuum to permit electric field lines.

The relevant Maxwell equation for electrostatics is:

This equation states that the divergence of the electric field at a point is proportional to the local charge density.

Coulomb’s Law

Definition and Formula

Coulomb’s Law quantifies the electric force between two point charges. The force is:

• Directly proportional to the product of the magnitudes of the charges.

• Inversely proportional to the square of the distance between them.

The mathematical expression is:

• \(F\): Magnitude of the electric force

• \(q_1, q_2\): Magnitudes of the two point charges

• \(r\): Distance between the charges

• \(k\): Coulomb’s constant,

Direction: The force is repulsive for like charges and attractive for opposite charges.

Electric Field

Concept and Definition

The electric field is a vector field that describes the influence a charge exerts on other charges in the space around it. It is defined as the force per unit positive test charge:

• \(\vec{E}\): Electric field at a point

• \(\vec{F}_0\): Force experienced by a test charge

• \(q_0\): Magnitude of the test charge

The direction of \(\vec{E}\) is the direction of the force on a positive test charge.

Visualizing the Electric Field

• When two charges are present, each exerts a force on the other.

• If one charge is removed, the remaining charge modifies the properties of space, creating an electric field at every point.

• The electric field at a point can be measured by placing a small test charge at that point and measuring the force exerted on it.

Key Point: The electric field is a property of space due to the presence of electric charges, independent of the presence of a test charge.

Electric Field of a Point Charge

The electric field produced by a point charge \(q\) at a distance \(r\) from the charge is given by:

• \(\hat{r}\): Unit vector pointing from the source charge to the field point

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

Electric Field Lines

Definition and Properties

• An electric field line is an imaginary line whose tangent at any point gives the direction of the electric field vector at that point.

• The density of field lines indicates the strength of the electric field: closer lines mean a stronger field.

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

• Field lines never cross.

Examples of Field Line Patterns

• Single Point Charge: Radial lines pointing outward (positive) or inward (negative).

• Dipole: Lines emerge from the positive charge and terminate at the negative charge.

• Two Equal Positive Charges: Lines radiate outward from both charges and never intersect.

Superposition Principle

The total electric field at a point due to multiple charges is the vector sum of the fields produced by each charge individually:

Applications: Electric Dipoles

Definition and Properties

• An electric dipole consists of two equal and opposite charges separated by a small distance.

• The electric dipole moment is defined as , where \(q\) is the charge and \(\vec{d}\) is the displacement vector from negative to positive charge.

Example: Water Molecule

• The water molecule is electrically neutral overall, but the arrangement of atoms causes a separation of charge, making it an electric dipole.

• This property allows water to dissolve ionic compounds by attracting positive and negative ions.

Force and Torque on a Dipole

• In a uniform electric field, the net force on a dipole is zero, but a torque acts to align the dipole with the field.

• The torque \(\vec{\tau}\) on a dipole is given by:

• \(\vec{p}\): Electric dipole moment

• \(\vec{E}\): Electric field

Summary Table: Key Concepts

Additional info: The notes above expand on the brief points and images provided, adding academic context and definitions for clarity and completeness.