Electromagnetic potentials retarded

We could now proceed to substitute the second term of (3.133) for A in (3.142) but, as mentioned earlier, the second term of (3.133) is incorrect. It has been found that the correct form of the interaction between the momentum P and this part of the vector potential is obtained only if the retardation of the electromagnetic potentials is included. We do not go into the details here but simply quote the resulting contribution to (3.142),... 

An alternative explanation of the mechanism of spontaneous emission has been proposed by Wheeler and Feynman (1945) in a theory which includes both the retarded and advanced electromagnetic potentials introduced in section 2.6. The difference between the two theories is most evident if we consider a hypothetical universe containing a single... 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

As a next step we also need to specify the magnetic and retardation interactions experienced by an electron i and generated by all other electrons. In a first approximation retardation is neglected and we assume that electron i experiences the electromagnetic field immediately. For the scalar potential j,unret, and the vector potential A/ Unret created by electron j and felt by electron i the classical expression reads ... 

So far, we considered only the unretarded electromagnetic field. However, for the correct expression, we have to include the retardation of the vector potential due to the finite speed of light. We may obtain from Darwin s classical electromagnetic interaction energy expression (21) (correct up to 0(c 2)),... 

Finally, it is worth remarking that retarded van der Waals dispersion potentials between molecules in ground and excited electronic states may also be calculated [12,51] using the fluctuating moment method. Because dispersion forces arise from the perturbation induced by the zero-point energy associated with the vacuum electromagnetic field, the expectation value of... 

Coalescence frequency J depends on dimensionless parameters k, p, Sa, Sr, t, y, a. The parameter k characterizes relative sizes of interacting drops p is the viscosity ratio of drops and ambient liquid Sa and Sr are the forces of molecular attraction and electrostatic repulsion of drops r is the relative thickness of electric double layer, which depends, in particular, on concentration of electrolyte in ambient liquid y is the electromagnetic retardation of molecular interaction a is relative potential of surfaces of interacting drops. Let us estimate the values of these parameters. For hydrosols, the Hamaker constant is F 10 ° J. For viscosity and density of external liquid take m /s, 10 kg/m. ... 

It is clear that all retardation effects are now coded in the exponential function under the integral. The unretarded potentials are obtained once the exponential vanishes, which is the case if tOmn is set equal to zero as we will investigate below. Then, the total interaction energy expectation value can be generalized from Eq. (2.150), i.e., expressed as the 4-current of electron 1 in the electromagnetic 4-potential generated by electron 2,... 

The microscopic method, credited to Hamaker, came first and is based on pair-wise summation of the individual dispersion interaction between molecules. Casmir and Polder later supplemented this approach by including the correction for electromagnetic retardation. The molecular interaction potential used is typically represented by the following expression ... 

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