Table of Contents
- Introduction
- Electric Charge and Coulomb’s Law
- Electric Fields and Gauss’s Law
- Electric Potential and Energy
- Conductors and Dielectrics
- Magnetic Fields and Biot–Savart Law
- Ampère’s Law and Magnetic Materials
- Faraday’s Law of Induction
- Maxwell’s Equations
- Electromagnetic Waves
- The Poynting Vector and Energy Transport
- Electromagnetism and Special Relativity
- Applications of Electromagnetism
- Conclusion
1. Introduction
Electromagnetism is one of the four fundamental forces of nature and governs all phenomena involving electric charges and magnetic fields. This classical theory, consolidated in the 19th century by James Clerk Maxwell, laid the foundation for modern physics — from electrical engineering to quantum electrodynamics.
This review article captures the essential laws, concepts, and equations of classical electromagnetism, serving as a bridge to advanced topics in field theory and quantum physics.
2. Electric Charge and Coulomb’s Law
Electric charge is a fundamental conserved quantity. There are two types of charge — positive and negative — and like charges repel while unlike charges attract.
Coulomb’s law describes the force between two point charges:
\[
\vec{F} = \frac{1}{4\pi \varepsilon_0} \frac{q_1 q_2}{r^2} \hat{r}
\]
Where:
- \(\varepsilon_0\) is the vacuum permittivity
- \(r\) is the separation between charges
3. Electric Fields and Gauss’s Law
The electric field \( \vec{E} \) is the force per unit charge:
\[
\vec{E} = \frac{\vec{F}}{q}
\]
For a point charge:
\[
\vec{E} = \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2} \hat{r}
\]
Gauss’s law:
\[
\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\varepsilon_0}
\]
4. Electric Potential and Energy
Electric potential \( V \) relates to the electric field:
\[
\vec{E} = -\nabla V
\]
Potential energy of two charges:
\[
U = \frac{1}{4\pi \varepsilon_0} \frac{q_1 q_2}{r}
\]
5. Conductors and Dielectrics
Conductors allow free charge movement — the electric field inside is zero in electrostatics.
In conductors:
- \( \vec{E} = 0 \) inside
- Charges reside on surface
Dielectrics polarize under an external field, weakening the effective field. This modifies Coulomb’s law and introduces the electric displacement field \(\vec{D}\):
In dielectrics:
\[
\vec{D} = \varepsilon_0 \vec{E} + \vec{P}, \quad \text{or} \quad \vec{D} = \varepsilon \vec{E}
\]
Where \(\varepsilon = \varepsilon_0 \varepsilon_r\)
6. Magnetic Fields and Biot–Savart Law
A moving charge or current produces a magnetic field:
\[
\vec{B} = \frac{\mu_0}{4\pi} \int \frac{I\, d\vec{\ell} \times \hat{r}}{r^2}
\]
This is the Biot–Savart law, which resembles Coulomb’s law for magnetism.
7. Ampère’s Law and Magnetic Materials
Ampère’s law relates magnetic fields to electric currents:
\[
\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}
\]
For magnetic materials, we define:
\[
\vec{B} = \mu_0(\vec{H} + \vec{M})
\]
Where:
- \(\vec{H}\): magnetic field intensity
- \(\vec{M}\): magnetization
- \(\mu\): magnetic permeability
8. Faraday’s Law of Induction
Changing magnetic fields induce electric fields:
\[
\oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}
\]
This underlies transformers, electric generators, and electromagnetic braking.
9. Maxwell’s Equations
Maxwell unified electricity and magnetism into four equations:
- Gauss’s Law for Electricity:
\[
\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}
\] - Gauss’s Law for Magnetism:
\[
\nabla \cdot \vec{B} = 0
\] - Faraday’s Law:
\[
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
\] - Ampère–Maxwell Law:
\[
\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}
\]
Together, these govern all classical electromagnetic phenomena.
10. Electromagnetic Waves
From Maxwell’s equations, one derives the wave equation:
\[
\nabla^2 \vec{E} – \mu_0 \varepsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} = 0
\]
The solution describes electromagnetic waves traveling at:
\[
c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}
\]
Thus, light is an electromagnetic wave!
11. The Poynting Vector and Energy Transport
The Poynting vector \(\vec{S}\) describes the energy flux:
\[
\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}
\]
Energy density:
\[
u = \frac{1}{2} \left( \varepsilon_0 E^2 + \frac{B^2}{\mu_0} \right)
\]
12. Electromagnetism and Special Relativity
Maxwell’s equations are Lorentz invariant — they led Einstein to formulate special relativity.
Electric and magnetic fields are aspects of the same electromagnetic field tensor, and transform into each other depending on the observer’s frame.
In other words, electric and magnetic fields are components of the same field tensor in relativity. Maxwell’s equations are Lorentz invariant, leading Einstein to develop special relativity.
13. Applications of Electromagnetism
Electromagnetism underlies:
- Electrical circuits and power grids
- Wireless communication (radio, microwaves, optics)
- MRI and electromagnetic imaging
- Plasma physics and astrophysics
- Quantum electrodynamics (QED)
14. Conclusion
Electromagnetism is a pillar of classical and modern physics. Reviewing its core concepts — from fields and forces to Maxwell’s equations and wave propagation — is essential for deeper exploration of quantum field theory, relativity, and the Standard Model.
