Coulomb potential between two current loops embedded in a magnetic field, and the magnetic interaction between current loops in a simple plasma or electron gas. As seen from the Coulomb interaction between tubes of charge example, shown below, these more complicated geometries can lead to such exotic phenomena as fractional quantum numbers. Selected examples The Yukawa potential: The force between two nucleons in an atomic nucleus Consider the spin-0 Lagrangian density The equation of motion for this Lagrangian is the Klein–Gordon equation If we add a disturbance the probability amplitude becomes If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes With the amplitude in this form it can be seen that the propagator is the solution of From this it can be seen that The energy due to the static disturbances becomes (see ) with which is attractive and has a range of Yukawa proposed that this field describes the force between two nucleons in an atomic nucleus. It allowed him to predict both the range and the mass of the particle, now known as the pion, associated with this field. Electrostatics The Coulomb potential in a vacuum Consider the spin-1 Proca Lagrangian with a disturbance where charge is conserved and we choose the Lorenz gauge Moreover, we assume that there is only a time-like component to the disturbance. In ordinary language, this means that there is a charge at the points of disturbance, but there are no electric currents. If we follow the same procedure as we did with the Yukawa potential we find that which implies and This yields for the timelike propagator and which has the opposite sign to the Yukawa case. In the limit of zero photon mass, the Lagrangian reduces to the Lagrangian for electromagnetism Therefore the energy reduces to the potential energy for the Coulomb force and the coefficients and are proportional to the electric charge. Unlike the Yukawa case, like bodies, in this electrostatic case, repel each other. Coulomb potential in a simple plasma or electron gas Plasma waves The dispersion relation for plasma waves is where is the angular frequency of the wave, is the plasma frequency, is the magnitude of the electron charge, is the electron mass, is the electron temperature (Boltzmann's constant equal to one), and is a factor that varies with frequency from one to three. At high frequencies, on the order of