About polarisation vectors and tensors

Question

I don't understand the concept of tensors. Can anyone explain the relation between polarization vectors and tensors?

Answer 1

The components $P_i$ of your polarization vector $\vec P$ need not in general be proportional only to the corresponding component $E_i$ of your electric field $\vec E$. If $\alpha$ is a scalar then certainly $$ P_i=\alpha E_i $$ but you can imagine a situation where $$ P_i=\sum_{j}\alpha_{ij}E_j $$ i.e. the polarization in the direction $i$ is a (linear) functions of all three components of $\vec E$. In this case, $\alpha$ is properly a $3\times 3$ matrix $$ \alpha= \left(\begin{array}{ccc} \alpha_{11}&\alpha_{12}&\alpha_{13}\\ \alpha_{21}&\alpha_{22}&\alpha_{23}\\ \alpha_{31}&\alpha_{32}&\alpha_{33}\end{array}\right) $$ and so is thought of as a tensor.

This is similar to the inertia tensor $\boldsymbol{I}$ with components $I_{ij}$ in rigid rotational dynamics, which allow to connect angular momentum with angular velocity: $$ \boldsymbol{L}=\boldsymbol{I}\cdot \boldsymbol{\omega} $$

Answer 2

Polarization is a vector quantity which is related to the Electric field (also a vector quantity) by some equation. There is a quantity alpha in front of the vector electric field which is a tensor. So to understand what the equation means you should know the difference between a vector and a tensor.

Vectors and Tensors are some quantities which transform in a particular way, which is different from each other. If you have dealt with the mathematics of vectors then you might know that a vector can be represented as a column matrix where each element represent the components of a vector.

On the other hand tensors are represented in the form of a N x N matrix, depending on the dimension of the problem.

In the above case the tensor alpha is a 3 x 3 matrix. So if you multiply a 3x3 matrix (alpha) to a 3x1 matrix (E), then the result is a 3x1 matrix (P).

Answer 3

When the dielectric is isotropic then the quantity α is scalar and x component of polarization vector depends only on x component of electric field and similar results for y and z components. If the dielectric is anisotropic then x component of polarization vector depends on all x,y and z components of electric field and similar is the case for y and z components. In this condition polarization vector is related to electric field vector by taking nine components of the quantity α in 3D space as in this condition α isn't a scalar, it's a tensor of rank 2. In fact, tensor analysis is the generalization of vector analysis. All physical quantities are tensors. Scalar is a tensor of zero rank, vector is a tensor of first rank, diadic is a tensor of 2nd rank etc. You must remember tensorial form of physical equations are invariant under coordinate transformation.

Answer 4

I am going to direct you and any serious physicist to http://www.e-booksdirectory.com/details.php?ebook=2221 (Thorne and Blandford's brilliant view of classical physics via the Geometric Principle). Here, they introduce rank-2 tensors as bilinear functions with 2 slots, in which 2 vectors go and a scalar pops out, or 1 vector goes in and another vector pops out. While this may seem trite compared with the definition based on coordinate transformation, it is in fact simpler and more powerful.

So for you, ${\bf \alpha}({\mathrm \_},{\mathrm \_} )$ is a function that takes an electric field vector in one slot and returns a polarization vector.

I suspect this may not be helpful--it's a big subject. I cannot recommend anything Kip Thorne does enough--whether its a book or youtube lecture.