1. Electric Charge and Basic Properties

Electric charge is a fundamental property of matter. There are two types of charge: positive and negative. Like charges repel each other, while unlike charges attract. Charge is quantized, meaning it exists in discrete units, the smallest being the elementary charge of an electron or proton. Charge is also conserved; it cannot be created or destroyed, only transferred. Materials can be classified based on their ability to conduct charge: conductors (allow charge to flow easily) and insulators (resist charge flow). Charging can occur through friction, conduction, or induction.

2. Coulomb's Law and Electric Field

Coulomb's Law quantifies the force between two point charges. It states that the force is directly proportional to the product of the charges ($q_1, q_2$) and inversely proportional to the square of the distance ($r$) between them: $F = k \frac{|q_1 q_2|}{r^2}$, where $k$ is Coulomb's constant. The electric field ($\vec{E}$) at a point is defined as the force per unit positive test charge placed at that point ($\vec{E} = \vec{F}/q_0$). Electric field lines originate from positive charges and terminate on negative charges, indicating the direction of the force a positive charge would experience.

3. Electric Dipoles and Fields

An electric dipole consists of two equal and opposite charges separated by a small distance. The strength of a dipole is characterized by its dipole moment ($\vec{p}$), a vector pointing from the negative to the positive charge with magnitude $p = qd$. Dipoles create electric fields that have a specific pattern, with field lines originating from the positive charge and terminating on the negative charge. The torque experienced by a dipole in a uniform external electric field is given by $\vec{\tau} = \vec{p} \times \vec{E}$.

4. Gauss's Law and its Applications

Gauss's Law provides a powerful way to calculate the electric field, especially for charge distributions with high symmetry. It relates the electric flux ($\Phi_E$) through a closed surface to the net charge ($q_{\text{enc}}$) enclosed within that surface: $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\epsilon_0}$, where $\epsilon_0$ is the permittivity of free space. Gauss's Law simplifies electric field calculations for conductors, charged spheres, infinite lines of charge, and infinite planes of charge.

5. Electrostatic Potential and Energy

The electrostatic potential ($V$) at a point in an electric field is the work done per unit charge to move a positive test charge from infinity to that point. It is related to the electric field by $\vec{E} = -\nabla V$. The potential difference (voltage) between two points is the work done per unit charge to move a charge between those points. Electrostatic potential energy ($U$) is the energy a system of charges possesses due to their relative positions, calculated as $U = qV$ for a charge $q$ at a potential $V$. For a system of multiple charges, it's the sum of potential energies of all pairs.

6. Conductors, Dielectrics, and Capacitance

In electrostatic equilibrium, the electric field inside a conductor is zero, and any net charge resides on its surface. Dielectrics are insulating materials that can be polarized by an external electric field, reducing the field strength within them. Capacitance ($C$) is a measure of a conductor's ability to store electric charge, defined as the ratio of the charge ($Q$) on the conductor to the potential difference ($V$) across it: $C = Q/V$. Capacitors, often made of conductors separated by a dielectric, are crucial components in electronic circuits.

7. Additional: Energy Density in Electric Field

The energy stored in an electric field is distributed throughout the space occupied by the field. The energy density ($u_E$) is the energy stored per unit volume. For a vacuum, it is given by $u_E = \frac{1}{2}\epsilon_0 E^2$, where $E$ is the electric field strength. In a dielectric medium, it is $u_E = \frac{1}{2}\epsilon E^2$, where $\epsilon = \epsilon_r \epsilon_0$ is the permittivity of the medium and $\epsilon_r$ is the relative permittivity (dielectric constant). This concept is fundamental to understanding how electric fields store energy.