Maxwell's Equations Explained: Supplement to the History of Maxwell's Eq.

This paragraph introduces fundamental concepts of electricity with a focus on vectors and their multiplication methods, namely the dot product and cross product. It explains that scalars like mass have value but no direction, whereas vectors like weight have both. The paragraph uses the example of force and distance to illustrate how the dot product results in work done (a scalar), while the cross product results in torque (a vector). The importance of the direction in vector operations is highlighted, and a mnemonic for cross product signs is introduced using a triangle of X, Y, and Z.

The second paragraph delves into the DEL function, also known as the vector differential operator, which is used to calculate derivatives in multiple directions. It introduces the concepts of divergence and curl in relation to the DEL function. Divergence measures how a vector field spreads out from a point, while curl measures how much it curls around a point. The paragraph connects these concepts to Maxwell's equations, particularly Gauss's law, which relates the divergence of the electric field to the charge density and asserts that the divergence of the magnetic field is always zero, indicating magnetic fields can only form closed loops.

This paragraph provides a historical context for Maxwell's equations, starting with Coulomb's torsion balance experiment and his discovery of the electric force's dependency on charge size and distance. It discusses Faraday's experiments with conductive cages and his concept of lines of electric and magnetic force. The paragraph then describes Maxwell's approach to modeling these lines of force using the displacement field D and Gauss's method of imaginary closed surfaces. The relationship between the electric field E and the displacement field D is also explained through Maxwell's constant K and the permittivity constant epsilon.

The fourth paragraph discusses Ampere's law and its extension by Maxwell to include a changing electric field as a source of magnetic field curl. It recounts Ampere's experiments with spirals of wire that acted like magnets and his discovery of the force between two parallel currents. The paragraph also explains Maxwell's hypothesis that a changing electric field induces a magnetic field, which was a key insight leading to the concept of electromagnetic induction. This historical account leads up to Maxwell's formulation of Ampere's law with an additional term accounting for the displacement current.

This paragraph explores the implications of Maxwell's equations for the understanding of electromagnetism and light. It describes how Maxwell's laws support the idea that all magnetic fields are due to the action of electric charges, unifying the electric and magnetic forces. The paragraph also discusses the historical significance of Maxwell's work in predicting the existence of electromagnetic waves and their propagation at the speed of light, which was later experimentally confirmed by Heinrich Hertz. The connection between Maxwell's equations and Einstein's mass-energy equivalence principle is highlighted, emphasizing the far-reaching impact of Maxwell's theories.

The final paragraph discusses the practical applications of Maxwell's laws, particularly in the field of wireless communication. It mentions the discovery of radio waves by Heinrich Hertz and how these findings validated Maxwell's laws. The paragraph also touches on the broader impact of Maxwell's work, including its influence on Albert Einstein's theory of relativity and the mass-energy equivalence formula E=mc². The summary concludes with a reference to the importance of Maxwell's laws in the ongoing wireless revolution and their foundational role in modern physics.