## Explanation

### The Relationship between Electricity and Magnetism

Electricity and magnetism are deeply intertwined and together make up the phenomenon known as **electromagnetism**. This relationship is succinctly described by **Maxwell's Equations**, which are fundamental to understanding how electric and magnetic fields interact.

### Faraday's Law of Induction

One of the key principles that illustrate this relationship is **Faraday's Law of Induction**:

Where:

- $\mathcal{E}$ is the induced electromotive force (EMF)
- $\Phi_B$ is the magnetic flux through a surface

**Faraday's Law** demonstrates that a changing magnetic field can induce an electric current in a conductor.

### Ampère's Law

Another cornerstone of electromagnetism is **Ampère's Law**, which relates magnetic fields to the currents that produce them:

Where:

- $\mathbf{B}$ is the magnetic field
- $\mu_0$ is the permeability of free space
- $\mathbf{J}$ is the current density
- $\epsilon_0$ is the permittivity of free space
- $\mathbf{E}$ is the electric field

**Ampère's Law** shows that electric currents and changing electric fields can generate magnetic fields.

### Maxwell's Equations

The most comprehensive way to describe the relationship between electricity and magnetism is through **Maxwell's Equations**. Combined, these equations form the foundation of classical electromagnetism:

These equations clearly demonstrate the **interdependent nature** of electric and magnetic phenomena.

### Electromagnetic Waves

The interplay between electric and magnetic fields gives rise to electromagnetic waves. An oscillating electric field generates a magnetic field and vice versa, allowing these waves to propagate through space. The wave equation for electromagnetic waves in a vacuum is:

$\nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$Where:

- $\nabla^2$ is the Laplace operator
- $\mathbf{E}$ is the electric field
- $\mu_0$ is the permeability of free space
- $\epsilon_0$ is the permittivity of free space

This equation underscores the nature of light and other forms of electromagnetic radiation as being fundamentally linked to electricity and magnetism.

**In summary**, the intrinsic relationship between electricity and magnetism is one of the cornerstones of modern physics, encapsulated by Maxwell's Equations and illustrated through phenomena like electromagnetic waves.