Maxwell stress on a small dielectric sphere in a dielectric

Electrically induced normal pressure and tangential stress at the surface of a small dielectric sphere (or cavity) in a dielectric are calculated using the Minkowski, Einstein-Laub, Abraham, and Lorentz forms of the Maxwell stress tensor. Only the Lorentz tensor is in agreement with the following observations: (1) A spherical cavity in a dielectric transforms into a sharp-edge plate perpendicular to the electric field; (2) a liquid drop placed in a medium with a slightly lower refractive index is stretched along the electric field; and (3) there is a torque on a small birefringent sphere. These phenomena cannot be explained by the conventional theory using the Minkowski stress tensor. For example, the Minkowski stress tensor predicts lateral compression of a spherical cavity in a dielectric.

Figure 1

The surface forces calculated with the Minkowski (first column), Einstein-Laub (second column), and Lorentz (third column) stress tensors. The upper row of the theoretical figures (a–c) demonstrates action of a constant electric field on an air bulb in water. The fourth column shows the photographs of a drop of silicon oil in castor oil in constant electric fields [34]. The forces calculated for this experiment are presented in the bottom row of the theoretical figures (d–f). These figures are also for the pressure of light at the surface of an empty spherical cavity in a dielectric with ${\epsilon }_1=2.27$. In all cases, the electric-field vector is directed vertically.

Figure 2

Pressure of ${\mathrm{CO}}_2$ laser emission on a water droplet calculated with the Minkowski (a), Einstein-Laub (b), and Lorentz (c) stress tensors.

Figure 3

Forces at the surface of a small E44 liquid-crystal sphere in water calculated using the Abraham (a) and Lorentz (b) stress tensors.

Figure 4

Deformation of a spherical cavity in an elastic dielectric in a homogeneous electric field calculated with the Minkowski (a), Einstein-Laub (b), and Lorentz (c) stress tensors. The insets show the surfaces defined by Eq. (A1) with one nonzero coefficient ${\alpha }_2$ equal to $0.1$ (a), $-0.1$ (b), and $-0.2$ (c).