Kishan Sinha

The dot product between two vectors \(\textbf{A} = (A_{x},A_{y},A_{z})\) and \(\textbf{B} = (B_{x},B_{y},B_{z})\) yields \(\textbf{A} \cdot \textbf{B} = A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}\). Here, the subscripts \(x,y\) and \(z\) denote components of vectors \(A\) and \(B\) along \((x,y\) and \(z\) axes respectively. In cases where the number of dimensions exceed 3, it is more convenient to label dimensions as {\(1,2,3,...\)} rather than letters {\(x,y,z,...\)}. With this change in labeling method, the vectors \(\textbf{A}\) and \(\textbf{B}\) can be rewritten as \(\textbf{A} = (A_{1},A_{2},A_{3})\) and \(\textbf{B} = (B_{1},B_{2},B_{3})\); and the dot product of the two vectors is \(\textbf{A} \cdot \textbf{B} = A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}\). Thus, we label or index the three dimensions as {\(1,2,3\)} - a set of three numbers. We can rewrite the dot product as \[\textbf{A} \cdot \textbf{B} = A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3} = \sum_{i=1}^{3} \, A_{i}B_{i}\] This is a...

PREVIOUS: Einstein's summation convention The dot product of two vectors \(\textbf{A}\) and \(\textbf{B}\) can be written, using Einstein's convention, as \(\textbf{A} \cdot \textbf{B} = A_{i}B_{i}\). The cross-product, on the other hand, is written in terms of Levi-Cevita symbol. The \(i\) the component of \(\textbf{A} \times \textbf{B}\) is: $$ (\textbf{A} \times \textbf{B})_{i} = \sum_{j,k=1}^{3}\epsilon_{ijk} A_{j}B_{k} = \epsilon_{ijk} A_{j}B_{k} $$ Here, \(\epsilon_{ijk}\) is known as Levi-Cevita symbol or a permutation symbol. Note that the indices \(j\) and \(k\) are repeated or dummy, hence summed over, but the index \(i\) is free. Therefore, as discussed in the previous section, the term \(\epsilon_{ijk} A_{j}B_{k} \) is like the \(i\)th component of a vector. However, it is common to write \(\textbf{A} \times \textbf{B} = \epsilon_{ijk} A_{j}B_{k} \) for notational brevity; this doesn't create any confusion as the free index \(i\) makes it ob...

In this post I present examples of application of the method of index notation and Einstein's summation convention. I find the technique extremely useful especially when I don't have access to a book of formulae. As a student, I was taught that it's always best to be mathematically self-sufficient - meaning should the need arise, one should be able to derive formulae or construct an analytical framework from scratch using a few established axioms in mathematics. Such proficiency in mathematics enables one to focus on the subject that one needs analyzed rather than on mathematics which only provides tools to facilitate analysis. I find that this, unfortunately, is not a universal view. Below, I show how some of the most widely applied equations in physics can be processed or derived readily using the index notation method. I derive these equations every time I need them. I must have done so a million times. So, I find the process easy to use. However, I have a...

PREVIOUS: Dielectric in an electric field In 1880, two French scientists Jacques and Pierre Curie, were also brothers discovered that pressure applied to certain crystals such as quartz, tourmaline, Rochelle salts and cane sugar creates an electrical charge in that material. They referred to this strange phenomenon as the piezoelectric effect. Soon afterwards, they observed the inverse piezoelectric effect where application of electric voltage across these crystals caused mechanical deformation. They subsequently obtained enough data to prove quantitatively the complete reversibility of this effect. Note that all the crystals listed above are dielectrics (electrical insulators). In piezoelectric materials, the crystal structure is such that the mechanical properties of the crystal are couple to its electronic properties at microscopic level so that any change in one causes a corresponding change in the other. This may also be referred to as electro-elasto-mechanical ...

A charge \( +Q \) placed in free space sets up an electric field around itself. This electric field is described by the Coulomb’s law of electrostatics. But a charge in a free space is quite useless: in real life we are surrounded by things of all sorts, and if we really want to make any use of our knowledge of electrostatics, we'll need to learn to apply it in presence of other materials. Motivated by the need to make our knowledge useful, let’s make this system a bit more interesting, and close to reality, by introducing a dielectric (insulator) material around the charge, as shown in the figure below. The region \(B\) is dielectric, whereas regions \(A\) and \(C\) are empty. The questions that naturally arise are: a) How do the dielectric and the charge \(+Q\) interact? b) Do properties of the dielectric subjected to the electric field from \(+Q\) remain the same or change? b) Does the presence of dielectric affect the way the electric field is setup in space...