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Optical Communication Mk Raina Ebook 11
Optical Fiber Communication. Chapter-2. Ray Theory Transmission in Optical Fiber.. Super Fibre Optics 2nd Edition by M. K. Dandage.Q:
Theoretical question about Biot-Savart law in $3$ dimensional space
According to Biot-Savart law, $
abla \times \vec{A} = \frac{\vec{j} \times \vec{r}}{r^3}$, when considering a circle around point A. I wonder in $3$ dimensional space: if we have a circle around point A in $3$ dimensional space, how is $\frac{\vec{j} \times \vec{r}}{r^3}$ determined? Or are there $3$ kind of forces (i.e. $\vec{j} \times \vec{r}$, $\vec{j} \times \vec{n}$, $\vec{n} \times \vec{r}$ in $3$ dimensional space?)
A:
There are three types of magnetic fields: solenoidal (with $
abla\times{\bf A}=0$), irrotational (i.e. with $
abla\cdot{\bf A}=0$) and potential fields (i.e. $curl{\bf A}+
abla\Phi=0$). The latter are the most important for magnetic levitation, so it is mostly about potential fields that we are interested in, for example for writing a vector field as the curl of a vector field ${\bf a}$.
Applying this to your problem, you’ll note that (in the Cartesian basis $e_1,e_2,e_3$) the vector field of a circle has both $
abla\times{\bf A}=0$ and $
abla\cdot{\bf A}=0$, so if the current is $\vec{j}=\text{constant}$, then ${\bf A}=\frac{\vec{j}\times{\bf r}}{r^3}$ is a potential field with potential function $\Phi(r)=\frac{j^2}{2}\ln\,r$.
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Raina Engineering College
Professor #61, V.M. Comenian University of Mysore.
Publications
Murali K Raina, A K Tiwari, A K. Tiwari, K Kumar, A Murali, P N Sampath, H. K. Balakrishnan, K.K. Raina and A. Ray Chaudhuri, “On the synthesis of ternary CdTaO3-δ:Mn4+/Mn3+,” Synthetic Metals 168, pp. 758–760, 2010. (Published on-line 3 May 2010).
Murali K Raina, Vikram Krishna, Mani K. Vijayakrishna, Annul K. Sajila, “Faceted cubic α-NaYF4:Yb3+,Er3+ UCNP prepared by aqueous co-precipitation method,” Nuovo Cimento C 36, pp. 526–531, 2013. (Published on-line 1 August 2013).
Murali K Raina, N R C Raju and K J Sree Bhushan, “EPR study of metal centered neutral atoms in NiO and NiS under VUV excitation,” Applied Physics Letters 89, pp. 29 (15 March, 2011).
Murali K Raina, Vikram Krishna, Mani K. Vijayakrishna, Annul K. Sajila, “Faceted cubic α-NaYF4:Yb3+,Er3+ UCNP prepared by aqueous co-precipitation method,” Applied Physics Letters 88, pp. 29 (15 March, 2011).
Murali K Raina, Vikram Krishna, Mani K. Vijayakrishna, Annul K. Sajila, “Faceted cubic α-NaYF4:Yb3+,Er3+ UCNP prepared by aqueous co-precipitation method,” Optics Express 19, pp. 892–896, 2011. (Published on-line 26 July 2011).
Murali K Raina, Vikram Krishna, Mani K. Vijayakrishna, Annul K. Sajila, “Influence of size and its distribution of the nanoparticles on the performance of Eu3+
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