Ritz ballistic theory is a theory in physics, first published in 1908 by Swiss physicist Walther Ritz. In 1908, Ritz published Recherches critiques sur l'Électrodynamique générale,[1] [2] a lengthy criticism of Maxwell-Lorentz electromagnetic theory, in which he contended that the theory's connection with the luminiferous aether (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, a theory competing with the special theory of relativity. The equation relates the force between two charged particles with a radial separation r relative velocity v and relative acceleration a, where k is an undetermined parameter from the general form of Ampere's force law as proposed by Maxwell. The equation obeys Newton's third law and forms the basis of Ritz's electrodynamics.
F=
q1q2 | \left[\left[1+ | |
4\pi\epsilon0r2 |
3-k | \left( | |
4 |
v | |
c |
\right)2-
3(1-k) | \left( | |
4 |
v ⋅ r | |
c2 |
\right)2-
r | |
2c2 |
(a ⋅ r)\right]
r | |
r |
-
k+1 | |
2c2 |
(v ⋅ r)v-
r | |
c2 |
(a)\right]
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 (
D
Ux
Ur
ax
Fx=eD\left[A1cos(\rhox)+B1
UxUr | |
c2 |
+C1
\rhoax | |
c2 |
\right]
where the coefficients
A1
B1
C1
u2/c2
2 | |
u | |
\rho |
/c2
X+x(t')=X'+x'(t')-(t-t')v'x
Developing the terms in the force equation, we find that the density of particles is given by
D\alpha
dt'e'dS | |
\rho2 |
=-
e'\partial\rho | |
c\rho2\partialn |
dSdn
The tangent plane of the shell of emitted particles in the stationary coordinate is given by the Jacobian of the transformation from
X'
X
\partial\rho | |
\partialn |
=
\partial(XYZ) | |
\partial(X'Y'Z') |
=
ae' | |
\rho2 |
\left(1+
\rhoa'\rho | |
c2 |
\right)
We can also develop expressions for the retarded radius
\rho
U\rho<\rho>
\rho=r\left(1+
ra'r | |
c2 |
\right)1/2
\rhox=rx+
r2a'x | |
2c2 |
U\rho=vr-v'r+
ra'r | |
c |
With these substitutions, we find that the force equation is now
Fx=
ee' | \left(1+ | |
r2 |
ra'r | |
c2 |
\right)\left[Acos(rx)\left(1-
3ra'r | |
2c2 |
\right)+A\left(
ra'x | \right)-B\left( | |
2c2 |
uxur | \right)-C\left( | |
c2 |
ra'x | |
c2 |
\right)\right]
Next we develop the series representations of the coefficients
A=\alpha0+\alpha1
u2 | |
c2 |
+\alpha2
| |||||||
c2 |
+...
B=\beta0+\beta1
u2 | |
c2 |
+\beta2
| |||||||
c2 |
+...
C=\gamma0+\gamma1
u2 | |
c2 |
+\gamma2
| |||||||
c2 |
+...
With these substitutions, the force equation becomes
Fx=
ee' | |
r2 |
\left[\left(\alpha0+\alpha1
| |||||||
c2 |
+\alpha2
| |||||||
c2 |
\right)cos(rx)-\beta0
uxur | |
c2 |
-\alpha0
ra'r | |
2c2 |
+\left(
ra'x | |
2c2 |
\right)(\alpha0-2\gamma0)\right]
Since the equation must reduce to the Coulomb force law when the relative velocities are zero, we immediately know that
\alpha0=1
2\gamma0-1=1
\gamma0=1
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
d2Fx=\sumi,j
deidej' | |
r2 |
\left[\left(1+\alpha1
| |||||||
c2 |
+\alpha2
| |||||||
c2 |
\right)cos(rx)-\beta0
uxur | |
c2 |
-\alpha0
ra'r | |
2c2 |
+
ra'x | |
2c2 |
\right]
Consider the diagram on the right, and note that
dqv=Idl
\sumi,jdeidej'=0
\sumi,jdeidej'
2 | |
u | |
x |
=-2dqdq'wxw'x
=-2II'dsds'cos\epsilon
\sumi,jdeidej'
2 | |
u | |
r |
=-2dqdq'wrw'r
=-2II'dsds'cos(rds)cos(rds)
\sumi,jdeidej'uxur=-dqdq'(wxw'r+w'xwr)
=-II'dsds'\left[cos(xds)cos(rds)+cos(rds)cos(xds')\right]
\sumi,jdeidej'a'r=0
\sumi,jdeidej'a'x=0
Plugging these expressions into Ritz's equation, we obtain the following
d2Fx=
II'dsds' | |
r2 |
\left[\left[2\alpha1cos\epsilon+2\alpha2cos(rds)cos(rds')\right]cos(rx)-\beta0cos(rds')cos(xds)-\beta0cos(rds)cos(xds')\right]
Comparing to the original expression for Ampere's force law
d2Fx=-
II'dsds' | |
2r2 |
\left[\left[(3-k)cos\epsilon-3(1-k)cos(rds)cos(rds')\right]cos(rx)-(1+k)cos(rds')cos(xds)-(1+k)cos(rds)cos(xds')\right]
we obtain the coefficients in Ritz's equation
\alpha1=
3-k | |
4 |
\alpha2=-
3(1-k) | |
4 |
\beta0=
1+k | |
2 |
From this we obtain the full expression of Ritz's electrodynamic equation with one unknown
F=
q1q2 | \left[\left[1+ | |
4\pi\epsilon0r2 |
3-k | \left( | |
4 |
v | |
c |
\right)2-
3(1-k) | \left( | |
4 |
v ⋅ r | |
c2 |
\right)2-
r | |
2c2 |
(a ⋅ r)\right]
r | |
r |
-
k+1 | |
2c2 |
(v ⋅ r)v-
r | |
c2 |
(a)\right]
In a footnote at the end of Ritz's section on Gravitation ([4] English translation) the editor says, "Ritz used k = 6.4 to reconcile his formula (to calculate the angle of advancement of perihelion of planets per century) with the observed anomaly for Mercury (41") however recent data give 43.1", which leads to k = 7. Substituting this result into Ritz's formula yields exactly the general relativity formula." Using this same integer value for k in Ritz's electrodynamic equation we get:
F=
q1q2 | |
4\pi\epsilon0r2 |
\left[\left[1-\left(
v | |
c |
\right)2+4.5\left(
v ⋅ r | |
c2 |
\right)2-
r | |
2c2 |
(a ⋅ r)\right]
r | |
r |
-
4 | |
c2 |
(v ⋅ r)v-
r | |
c2 |
(a)\right]