Matter is composed of atoms, which contain positively charged protons and negatively charged electrons. The fundamental unit of charge ($e$) is $1.602 \times 10^{-19}$ Coulombs (C).

Charge ($q$) is a conserved property. The total charge of an isolated system remains constant. Like charges repel; opposite charges attract.

The fundamental force between two point charges ($q_1$ and $q_2$) separated by a distance ($r$) is given by Coulomb's Law. It is mathematically identical in form to Newton's Law of Universal Gravitation, but the force can be attractive or repulsive.

Instead of thinking of charges exerting forces directly on one another across empty space, we introduce the concept of an electric field. A charge creates an electric field in the space around it, and another charge placed in that field feels a force.

Gauss's Law is a powerful alternative to Coulomb's law for calculating electric fields of symmetric charge distributions (like spheres, cylinders, or infinite planes). It relates the electric flux ($\Phi_E$) passing through a closed "Gaussian" surface to the total charge enclosed within that surface ($q_{enc}$).

Just as a mass in a gravitational field has gravitational potential energy, a charge in an electric field has electric potential energy ($U$). Because the electrostatic force is conservative, we can define a potential energy function.

It is often more useful to talk about energy per unit charge, which is the electric potential (commonly called Voltage).

The difference in potential between two points ($\Delta V$) is the voltage. It represents the work done per unit charge by an external agent moving a positive test charge between those points.

A capacitor is a device (usually two conducting plates separated by an insulator) that stores electric charge and potential energy.

The energy stored in a capacitor is $U = \frac{1}{2} C (\Delta V)^2$.

When charges move continuously, they form an electric current.

By convention, the direction of current is the direction positive charges would flow (opposite to the actual flow of electrons).

To drive a current through a conductor, a potential difference ($\Delta V$) is required. Most materials resist this flow.

Resistance depends on the material's resistivity ($\rho$) and geometry: $R = \rho L / A$.

The rate at which energy is dissipated in a resistor (power) is $P = I \Delta V = I^2 R = (\Delta V)^2 / R$.

For analyzing complex circuits, we use two rules based on conservation laws:

Moving charges (currents) create magnetic fields ($\vec{B}$). Magnetic fields, in turn, exert forces on moving charges.

Magnetic fields are created by currents. The Biot-Savart Law and Ampere's Law allow us to calculate these fields.

Here, $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$).

A changing magnetic field can induce an electric current in a conductor. This is the principle behind all electrical generators.

The negative sign represents Lenz's Law, which states that the induced current will flow in a direction such that its own magnetic field opposes the change in flux that caused it. This is a consequence of the conservation of energy.

In DC circuits, components can be connected in two primary configurations, which dictate how voltage and current behave across them.

When an insulating material (a dielectric) is placed between the plates of a capacitor, it increases the capacitance of the device by a dimensionless factor called the dielectric constant ($\kappa$).

The molecules in the dielectric become polarized by the external electric field, creating an opposing internal electric field. This reduces the overall electric field between the plates, allowing the capacitor to store more charge for the same applied voltage.