Electric Charges, Forces, and Fields

Electric Charge

Electric charge is a fundamental property of matter responsible for electric phenomena. The effects of electric charge were first observed as static electricity, such as when an amber rod, after being rubbed with fur, attracts small objects.

• Two Types of Charge: There are two types of electric charge: positive and negative. Like charges repel, and opposite charges attract.

• Elementary Charge: All electrons have the same charge, denoted as e. The charge of a proton is equal in magnitude but opposite in sign to that of an electron.

• SI Unit: The SI unit of charge is the coulomb (C).

• Quantization: Electric charge is quantized in units of e ($e = 1.60 \times 10^{-19}$ C).

• Conservation: The total electric charge in the universe is conserved.

• Ions: Atoms that lose electrons become positive ions; those that gain electrons become negative ions.

• Polarization: Some materials can become polarized, meaning their atoms rotate in response to an external charge, allowing a charged object to attract a neutral one.

Insulators and Conductors

Materials can be classified based on their ability to conduct electric charge.

• Conductors: Materials (usually metals) whose conduction electrons are free to move throughout the material.

• Insulators: Materials (usually nonmetals) whose electrons seldom move from atom to atom.

• Semiconductors: Materials with properties intermediate between conductors and insulators; their conductivity can change with chemical composition or exposure to light (photoconductivity).

• Charge Distribution: Excess charge on a conductor resides on its surface.

Coulomb’s Law

Coulomb’s law describes the force between two point charges.

• Formula: The magnitude of the force between two point charges is given by: $F = k \frac{|q_1 q_2|}{r^2}$ where $k = 8.99 \times 10^9\ \mathrm{N\,m^2/C^2}$, $q_1$ and $q_2$ are the charges, and $r$ is the distance between them.

• Direction: The force acts along the line connecting the charges. It is attractive if the charges are opposite and repulsive if they are alike.

• Action-Reaction: The forces on the two charges are equal in magnitude and opposite in direction (Newton’s third law).

• Superposition Principle: For multiple charges, the net force is the vector sum of the forces from each charge.

• Spherical Distributions: Coulomb’s law applies to spherically symmetric charge distributions, analogous to gravitational forces.

The Electric Field

The electric field is a vector field that describes the force per unit charge at each point in space.

• Definition: $E = \frac{F}{q_0}$, where $q_0$ is a test charge.

• Units: Newtons per coulomb (N/C).

• Force on a Charge: $F = qE$

• Direction: The force is in the direction of the field for a positive charge and opposite for a negative charge.

• Point Charge Field: The electric field of a point charge is: $E = k \frac{|q|}{r^2}$ It points radially away from positive charges and toward negative charges.

• Superposition: Electric fields from multiple charges add as vectors.

Electric Field Lines

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

• Field lines point in the direction of the electric field vector at every point.

• They start at positive charges (or infinity) and end at negative charges (or infinity).

• The density of lines indicates the strength of the field (more lines = stronger field).

• Field lines never cross.

• Combinations of charges produce characteristic field line patterns, with unique points where the field may be zero.

• Parallel-plate capacitors produce uniform electric fields between the plates.

Shielding and Charging by Induction

Conductors exhibit unique behaviors in electric fields due to the mobility of their charges.

• Surface Charge: Excess charge on a conductor resides on its surface.

• Field Inside Conductor: The electric field inside a conductor is zero when charges are at rest.

• Field Orientation: The electric field is always perpendicular to the surface of a conductor.

• Curvature Effect: The electric field is stronger where the surface is more sharply curved.

• Charging by Induction: A conductor can be charged by induction if grounded, allowing like charges to leave and leaving only excess charge when isolated.

Electric Flux and Gauss’s Law

Electric flux quantifies the amount of electric field passing through a surface, and Gauss’s law relates this flux to the charge enclosed.

• Electric Flux: $\Phi = EA \cos \theta$ Where $E$ is the electric field, $A$ is the area, and $\theta$ is the angle between the field and the normal to the surface.

• Units: $\mathrm{N\,m^2/C}$

• Gauss’s Law: $\Phi = \frac{q_{\text{encl}}}{\varepsilon_0}$ Where $q_{\text{encl}}$ is the charge enclosed by the surface, and $\varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2/(N\,m^2)}$ is the permittivity of free space.

• Applications: Gauss’s law is especially useful for finding electric fields in systems with high symmetry (spherical, cylindrical, planar).

Summary Table: Key Concepts of Chapter 19