Torque on a Moving Electric/Magnetic Dipole

Abstract

Abstract-We derive an expression for the torque exerted on an electric/magnetic dipole moving in an electromagnetic field, which contains two new velocity-dependent terms that to our knowledge were not reported before. A physical meaning of various torque components is discussed in terms of Lorentz force law and hidden momentum contribution.

Origin of the spin–orbit interaction

“…Also, the external -field acting on the electron is ( , ) = ( ), given by Eq. (20). Assuming the electron arrives at ( , , ) = ( , 0, 0) at = 0, in the immediate vicinity of = 0, the free charge-density, polarization, and magnetization may be written as…”

“…(28)(29)(30)(31) and with the aid of Eq. (20), we calculate the Einstein-Laub force exerted by the nucleus on the moving electron, as follows:…”

“…In the present problem, there is no external magnetic field in the rest-frame of the nucleus, that is, H r t ( , ) 0. = Also, the external E-field acting on the electron is E r E r t ( , ) ( ), n = given by equation (20). Assuming the electron arrives at x y z r ( , , ) ( , 0, 0) = at t 0, = in the immediate vicinity of t 0, = the free charge-density, polarization, and magnetization may be written as…”

“…Over the years, several authors have attempted to simplify Thomas's argument and presented alternative routes to arriving at the desired ½ factor [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Some even questioned the validity of Thomas's reasoning (though not that of his final result).…”

“…The search for the origin of the Thomas ½ factor in the rest-frame of the nucleus (including an examination of the role of the induced electric dipole moment ) P has a long and distinguished history. References [12][13][14][15][16][17][18][19][20][21] as well as those cited therein provide a good starting point for delving into the subject. In this connection, previous efforts have typically aimed at clarifying the spin dynamics in the rest-frame of the nucleus, thus helping to relate the behavior of the magnetic moment μ of the electron in its own rest-frame to that in the rest-frame of the nucleus.…”