text: In cases where the diameter of a trapped particle is significantly smaller than the wavelength of light, the conditions for Rayleigh scattering are satisfied and the particle can be treated as a point dipole in an inhomogeneous electromagnetic field. The force applied on a single charge in an electromagnetic field is known as the Lorentz force,
  The force on the dipole can be calculated by substituting two terms for the electric field in the equation above, one for each charge. The polarization of a dipole is  where  is the distance between the two charges. For a point dipole, the distance is infinitesimal,  Taking into account that the two charges have opposite signs, the force takes the form
  Notice that the  cancel out. Multiplying through by the charge, , converts position, , into polarization, ,
  where in the second equality, it has been assumed that the dielectric particle is linear (i.e. ).
  In the final steps, two equalities will be used: (1) a vector analysis equality, (2) Faraday's law of induction.
  First, the vector equality will be inserted for the first term in the force equation above. Maxwell's equation will be substituted in for the second term in the vector equality. Then the two terms which contain time derivatives can be combined into a single term.
  The second term in the last equality is the time derivative of a quantity that is related through a multiplicative constant to the Poynting vector, which describes the power per unit area passing through a surface. Since the power of the laser is constant when sampling over frequencies much longer than the frequency of the laser's light ~10 Hz, the derivative of this term averages to zero and the force can be written as
  where in the second part we have included the induced dipole moment (in MKS units) of a spherical dielectric particle: , where  is the particle radius,  is the index of refraction of the particle and  is the relative refractive index between the particle and the medium. The square of the magnitude of the electric field is equal to the intensity of the beam as a function of position. Therefore, the result indicates that the force on the dielectric particle, when treated as a point dipole, is proportional to the gradient along the intensity of the beam. In other words, the gradient force described here tends to attract the particle to the region of highest intensity. In reality, the scattering force of the light works against the gradient force in the axial direction of the trap, resulting in an equilibrium position that is displaced slightly downstream of the intensity maximum. Under the Rayleigh approximation, we can also write the scattering force as
  Since the scattering is isotropic, the net momentum is transferred in the forward direction. On the quantum level, we picture the gradient force as forward Rayleigh scattering in which identical photons are created and annihilated concurrently, while in the scattering (radiation) force the incident photons travel in the same direction and ‘scatter’ isotropically. By conservation of momentum, the particle must accumulate the photons' original momenta, causing a forward force in the latter.