Ritz's Equation
Encyclopedia
In 1908, Walter Ritz
Walter Ritz
Walther Ritz was a Swiss theoretical physicist.His father, Raphael Ritz, a native of Valais, was a well-known landscape and interior scenes artist. His mother was the daughter of the engineer Noerdlinger of Tübingen. Ritz studied in Zurich and Göttingen...

 published Recherches critiques sur l'Électrodynamique Générale, a lengthy criticism of Maxwell-Lorentz electromagnetic theory
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

, in which he contended that the theory's connection with the luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....

 (see Lorentz ether theory) made it "essentially inappropriate to express the comprehensive laws for the propagation of electrodynamic actions." Ritz proposed a new equation, derived from the principles of the ballistic theory of electromagnetic waves. This equation forms the basis of Ritz's electrodynamics.

Derivation of Ritz's Equation

On the assumption of an emission theory, the force acting between two moving charges should depend on the density of the messenger particles emitted by the charges (), the radial distance between the charges (ρ), the velocity of the emission relative to the receiver, ( and for the x and r components, respectively), and the acceleration of the particles relative to each other (). This gives us an equation of the form :
.


where the coefficients , and are independent of the coordinate system and are functions of and . The stationary coordinates of the observer relate to the moving frame of the charge as follows


Developing the terms in the force equation, we find that the density of particles is given by


The tangent plane of the shell of emitted particles in the stationary coordinate is given by the Jacobian of the transformation from to :


We can also develop expressions for the retarded radius and velocity using Taylor series expansions


With this substitutions, we find that the force equation is now


Next we develop the series representations of the coefficients


With these substitutions, the force equation is now


Since the equation must reduce to the Coulomb force law when the relative velocities are zero, we immediately know that . Furthermore, to obtain the correct expression for electromagnetic mass, we may deduce that or .
To determine the other coefficients, we consider the force on a linear circuit using Ritz's expression, and compare the terms with the general form of Ampere's law. The second derivative of Ritz's equation is

Consider the diagram on the right, and note that ,







Plugging these expressions into Ritz's equation, we obtain the following


Comparing to the original expression for Ampere's force law
Ampère's force law
In magnetostatics, the force of attraction or repulsion between two current-carrying wires is often called Ampère's force law...




we obtain the coefficients in Ritz's equation


From this we obtain the full expression of Ritz's electrodynamic equation with one unknown
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