The force density within a dielectric medium

Equation (3.67) was derived by considering a virtual process in which true charges are added to a system of charges and dielectrics which are held fixed, so that no mechanical work is done against physical displacements. Let us now consider a different virtual process in which the physical coordinates of the charges and dielectric are given a virtual displacement $\delta{\bfm r}$ at each point in space, but no free charges are added to the system. Since we are dealing with a conservative system, the energy expression (3.67) can still be employed, despite the fact that it was derived in terms of another virtual process. The variation in the total electrostatic energy $\delta U$ when the system undergoes a virtual displacement $\delta{\bfm r}$ is related to the electrostatic force density ${\bfm f}$ acting within the dielectric medium via

$$\delta U = - \int {\bfm f}\!\cdot\! \delta {\bfm r}\,d^3{\bfm r}.$$ (490)

If the medium is moving with a velocity field ${\bfm u}$ then the rate at which electrostatic energy is drained from the ${\bfm E}$ and ${\bfm D}$ fields is given by

$$\frac{dU}{dt} = - \int {\bfm f}\!\cdot\! {\bfm u}\,d^3{\bfm r}.$$ (491)

Let us now consider the energy increment due to both a change $\delta\rho_f$ in the free charge distribution and a change $\delta\epsilon$ in the dielectric constant, caused by the displacements. From Eq. (3.67)

$$\delta U = \frac{1}{2\epsilon_0}\int \left[D^2\delta(1/\epsi... ...\bfm D}\!\cdot\!\delta {\bfm D}/\epsilon\right]\, d^3{\bfm r},$$ (492)

or

$$\delta U = -\frac{\epsilon_0}{2}\int E^2\,\delta\epsilon \, d^3{\bfm r} + \int {\bfm E} \!\cdot\!\delta{\bfm D}\,d^3{\bfm r}.$$ (493)

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

$$\int{\bfm E}\!\cdot\! \delta{\bfm D}\,d^3{\bfm r} = - \int\n... ... {\bfm D}\,d^3{\bfm r} = \int \phi\,\delta\rho_f\,d^3{\bfm r},$$ (494)

where surface terms have been neglected. Thus, Eq. (3.72) implies that

$$\frac{dU}{dt} = \int\left(\phi\,\frac{\partial\rho_f}{\parti... ...E^2\, \frac{\partial\epsilon}{\partial t}\right)\,d^3{\bfm r}.$$ (495)

In order to arrive at an expression for the force density ${\bfm f}$ we need to express the time derivatives $\partial\rho/\partial t$ and $\partial\epsilon/\partial t$ in terms of the velocity field ${\bfm u}$. This can be achieved by adopting a dielectric equation of state; i.e., a relation which gives the dependence of the dielectric constant $\epsilon$ on the mass density $\rho_m$. Let us assume that $\epsilon(\rho_m)$ is a known function. It follows that

$$\frac{D\epsilon}{Dt} = \frac{d\epsilon}{d\rho_m} \frac{D\rho_m}{Dt},$$ (496)

where

$$\frac{D}{Dt} \equiv \frac{\partial}{\partial t} + {\bfm u}\!\cdot\!\nabla$$ (497)

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

$$\frac{\partial \rho_m}{\partial t} + \nabla\!\cdot\!(\rho_m {\bfm u}) = 0,$$ (498)

which implies that

$$\frac{D\rho_m}{Dt} = - \rho_m \nabla\!\cdot\!{\bfm u}.$$ (499)

It follows that

$$\frac{\partial \epsilon}{\partial t} = - \frac{d\epsilon}{d\... ..._m \nabla\!\cdot\!{\bfm u} - {\bfm u} \!\cdot\!\nabla\epsilon.$$ (500)

The conservation equation for the free charges is written

$$\frac{\partial \rho_f}{\partial t} + \nabla\!\cdot\!(\rho_f {\bfm u}) = 0.$$ (501)

Thus, we can express Eq. (3.74) in the form

$$\frac{dU}{dt} = \int\left[-\phi\,\nabla\!\cdot\!(\rho_f{\bfm... ...,\nabla\epsilon\right) \!\cdot\! {\bfm u}\right]\,d^3{\bfm r}.$$ (502)

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

$$-\int\phi\,\nabla\!\cdot\!(\rho_f{\bfm u}) \,d^3{\bfm r} = \int \rho_f \,\nabla\phi\!\cdot\! {\bfm u}\,d^3{\bfm r}.$$ (503)

Likewise,

$$\int\frac{\epsilon_0}{2}\,E^2\,\frac{d\epsilon}{d\rho_m} \,\... ...silon}{d\rho_m}\,\rho_m\right)\!\cdot\! {\bfm u}\,d^3{\bfm r}.$$ (504)

Thus, Eq. (3.81) becomes

$$\frac{dU}{dt} = \int\left[-\rho_f {\bfm E} +\frac{\epsilon_0... ...}{d\rho_m}\,\rho_m\right) \right]\cdot {\bfm u}\, d^3{\bfm r}.$$ (505)

Comparing with Eq. (3.70), we see that the force density inside the dielectric is given by

$${\bfm f} = \rho_f {\bfm E} - \frac{\epsilon_0}{2} \,E^2\,\na... ...\,\nabla\!\left(E^2 \frac{d\epsilon}{d\rho_m}\,\rho_m \right).$$ (506)

The first term in the above equation is the standard electrostatic force density. The second term represents a force which appears whenever an inhomogeneous dielectric is placed in an electric field. The last term, 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 depends explicitly on the dielectric equation of state of the material, through $d\epsilon/d\rho_m$. The electrostriction term gives zero net force acting on any finite region of dielectric if 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, since in the calculation of the total forces acting on dielectric bodies it usually does not contribute. Note, however, that if the electrostriction term is omitted an incorrect pressure variation within the dielectric is obtained, even though the total force is given correctly.