By James N. Pitts, etc.
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ELECTRIC STRESS TENSOR 29 where the subscripts and indicates upward and downward orientations respectively. Here t is any vector tangent to the interface and it follows that the potentials on either side of the interface differ by at most a constant. If no work is done in transferring charge across the interface, the constant is zero. e. the stress that an external field exerts on a surface. We will see first the expression of the stress tensor in a homogeneous fluid, then illustrate its application to a sphere immersed in an uncharged dielectric.
Electric fields polarize matter in two ways: by orienting molecules with permanent dipoles and by deforming electron clouds of each molecule. The polarization vector P is related to the characteristics of individual dipoles by the relation: 26 CHAPTER 2. ELECTROSTATICS N represents the number of dipoles per unit volume, Q is the magnitude of the charge separated to produce the dipole, and d is a vector describing the average orientation of the dipole and the charge separation distance. 16 together, indicate that the volumetric polarization charge is also zero.
The vector field v(r, t) is free of sources meaning that it can be discussed with the help of its accompanying rotational field, defined by a vector A(r,t) related to v(r,t) Here A is an axial vector (or pseudovector) whereas v is a polar vector. Axial vectors show the particularity of being changed to their opposites upon inversion of the coordinate system, in contrast to polar vectors (this is the case of the magnetic field in Electromagnetism). 2 Velocity field around a sphere Due to the symmetry of a moving sphere of radius R, the velocity v of the fluid only depends on the distance variable r (the origin of the coordinate system is taken at the center of the sphere) and on the velocity of the sphere U, and so does the vector A.