Electromagnetism

Electromagnetism: Definition and Mathematical Representation

Introduction

Electromagnetism is a fundamental branch of physics that deals with the study of electric and magnetic fields and their interactions with matter. It is one of the four fundamental forces of nature and is described by the unified theory of electricity and magnetism, primarily governed by Maxwell’s equations.

Electromagnetic phenomena are the basis for many modern technologies including electricity, magnetism, radio waves, microwaves, and optics.

Key Concepts

1. Electric Field ($\overrightarrow{E}$)

The force per unit charge exerted on a test charge:

$\overrightarrow{E}=\frac{\overrightarrow{F}}{q}$

For a point charge:

$\overrightarrow{E}=\frac{1}{4\pi {\epsilon }_0}\cdot \frac{q}{r^2}\hat{r}$

Where:

• $q$ is the source charge,
• $r$ is the distance from the charge,
• ${\epsilon }_0$ is the vacuum permittivity.

2. Magnetic Field ($\overrightarrow{B}$)

The field that exerts a force on moving charges or currents. It is defined by:

$\overrightarrow{F}=q\overrightarrow{v}\times \overrightarrow{B}$

Where:

• $\overrightarrow{F}$ is the magnetic force,
• $\overrightarrow{v}$ is the velocity of the charge,
• $\overrightarrow{B}$ is the magnetic field.

3. Electromagnetic Waves

Electric and magnetic fields oscillate perpendicular to each other and to the direction of wave propagation.

Speed of electromagnetic waves in vacuum:

$c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$

Where:

• $c$ is the speed of light,
• ${\mu }_0$ is the vacuum permeability,
• ${\epsilon }_0$ is the vacuum permittivity.

Maxwell’s Equations

Maxwell’s equations summarize the behavior of electric and magnetic fields:

1. Gauss's Law (Electric):

$\mathrm{\nabla }\cdot \overrightarrow{E}=\frac{\rho }{{\epsilon }_0}$

2. Gauss's Law (Magnetic):

$\mathrm{\nabla }\cdot \overrightarrow{B}=0$

3. Faraday's Law of Induction:

$\mathrm{\nabla }\times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$

4. Ampère-Maxwell Law:

$\mathrm{\nabla }\times \overrightarrow{B}={\mu }_0\overrightarrow{J}+{\mu }_0{\epsilon }_0\frac{\partial \overrightarrow{E}}{\partial t}$

Where:

• $\overrightarrow{E}$ is the electric field,
• $\overrightarrow{B}$ is the magnetic field,
• $\rho$ is the charge density,
• $\overrightarrow{J}$ is the current density.

Lorentz Force Law

The total electromagnetic force on a charged particle:

$\overrightarrow{F}=q(\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B})$

Energy in Electromagnetism

• Electric potential energy:

$U=qV$

• Magnetic energy density:

$u_B=\frac{1}{2{\mu }_0}{B}^2$

• Poynting vector (energy flow per unit area):

$\overrightarrow{S}=\frac{1}{{\mu }_0}\overrightarrow{E}\times \overrightarrow{B}$

Applications

• Electric power generation and transmission
• Wireless communication (radio, Wi-Fi)
• Motors and transformers
• Optics and laser technology
• Electromagnetic radiation (light, X-rays)

See Also

• Electric Field
• Magnetic Field
• Maxwell's Equations
• Electromagnetic Wave
• Lorentz Force
• Ampère's Law
• Faraday's Law
• Gauss's Law