(568) |

If the medium is moving with a velocity field then the rate at which electrostatic energy is drained from the and fields is given by

Consider the energy increment due to a change, , in the free charge distribution, and a change, , in the dielectric constant, which are both assumed to be caused by the virtual displacement. From Equation (567),

(570) |

or

Here, the first term represents the energy increment due to the change in dielectric constant associated with the virtual displacement, whereas the second term corresponds to the energy increment caused by displacement of the free charges. The second term can be written

(572) |

where surface terms have been neglected. Thus, Equation (572) implies that

In order to arrive at an expression for the force density, , we need to express the time derivatives and in terms of the velocity field, . This can be achieved by adopting a dielectric equation of state: that is, a relation that specifies the dependence of the dielectric constant, , on the mass density, . Let us assume that is a known function. It follows that

(574) |

where

(575) |

is the total time derivative (i.e., the time derivative in a frame of reference that is locally co-moving with the dielectric.) The hydrodynamic equation of continuity of the dielectric is

(576) |

which implies that

(577) |

It follows that

(578) |

The conservation equation for the free charges is written

(579) |

Thus, we can express Equation (574) in the form

Integrating the first term by parts, and neglecting any surface contributions, we obtain

(581) |

Likewise,

(582) |

Thus, Equation (581) becomes

(583) |

Comparing with Equation (570), we deduce that the force density inside the dielectric is given by

The first term in the above equation is the standard electrostatic
force density (due to the presence of free charges). The second term represents a force that appears whenever
an inhomogeneous dielectric is placed in an electric field. The last
term, which is known as the *electrostriction* term, gives a force acting
on a dielectric in an inhomogeneous electric field. Note that the
magnitude of the electrostriction force density depends explicitly on the
dielectric equation of state of the material, through
.
The electrostriction term gives zero net force acting on any finite region
of dielectric, provided we can integrate over a large enough portion of the dielectric that its extremities lie in a field-free region. For this reason,
the term is frequently omitted, because in the calculation of the
total forces acting on dielectric bodies it usually makes no contribution.
Note, however, that if the electrostriction
term is omitted then an incorrect pressure
variation within the dielectric is obtained, even though the total force is
given correctly.