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andresordonez
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Hi, I'm reading this article (you may need to register to view it, the registration is free though).
http://www.sciencemag.org/content/312/5781/1780.full
(can I post a link to this article in Dropbox so that people reading this don't have to register without getting an infraction from the moderators??)
and I'm getting this:
<br /> \epsilon'_{r'} = \epsilon \frac{R_2}{R_2-R_1} (r'-R_1)^2 \sin(\theta')<br />
<br /> \epsilon'_{\theta'} = \epsilon \frac{R_2}{R_2-R_1} \sin(\theta')<br />
<br /> \epsilon'_{\phi'} = \epsilon \frac{R_2}{R_2-R_1} \sin(\theta')<br />
instead of equations (7) in Pendry's article:
<br /> \epsilon'_{r'} = \frac{R_2}{R_2-R_1} \frac{(r'-R_1)^2}{r'}<br />
<br /> \epsilon'_{\theta'} = \frac{R_2}{R_2-R_1}<br />
<br /> \epsilon'_{\phi'} = \frac{R_2}{R_2-R_1}<br />
The difference between these equations and the ones I get is not only the missing r' and the extra sin(\theta') but also the extra \epsilon
This is what I'm doing. The new coordinates are given by equations (6):
<br /> r^{\prime}=R_{1}+r\frac{\left(R_{2}-R_{1}\right)}{R_{2}}<br />
<br /> \theta^{\prime}=\theta<br />
<br /> \phi^{\prime}=\phi<br />
The permittivity transforms according to:
<br /> \epsilon_{r}^{\prime}=\epsilon\frac{Q_{\theta'}Q_{\phi'}}{Q_{r'}}<br />
<br /> \epsilon_{\theta}^{\prime}=\epsilon\frac{Q_{r'}Q_{\phi'}}{Q_{\theta'}}<br />
<br /> \epsilon_{\phi}^{\prime}=\epsilon\frac{Q_{r'}Q_{\theta'}}{Q_{\phi'}}<br />
where Q_{u} is given by:
<br /> Q_u^2 = \left(\frac{\partial x}{\partial u}\right)^2 + \left(\frac{\partial y}{\partial u}\right)^2 + \left(\frac{\partial z}{\partial u}\right)^2<br />
Then:
<br /> Q_{r^{\prime}}^{2}=\left(\frac{\partial x}{\partial r^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial r^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial r^{\prime}}\right)^{2}<br />
<br /> \frac{\partial x}{\partial r^{\prime}}=\frac{\partial x}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial x}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial x}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial x}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\sin\theta\cos\phi\frac{R_{2}}{R_{2}-R_{1}}=\sin\theta^{\prime}\cos\phi^{\prime}\frac{R_{2}}{R_{2}-R_{1}}<br />
<br /> \frac{\partial y}{\partial r^{\prime}}=\frac{\partial y}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial y}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial y}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial y}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\sin\theta\sin\phi\frac{R_{2}}{R_{2}-R_{1}}=\sin\theta^{\prime}\sin\phi^{\prime}\frac{R_{2}}{R_{2}-R_{1}}<br />
<br /> \frac{\partial z}{\partial r^{\prime}}=\frac{\partial z}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial z}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial z}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial z}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\cos\theta\frac{R_{2}}{R_{2}-R_{1}}=\cos\theta^{\prime}\frac{R_{2}}{R_{2}-R_{1}}<br />
<br /> Q_{r^{\prime}}^{2}=\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}<br />
<br /> Q_{\theta^{\prime}}^{2}=\left(\frac{\partial x}{\partial\theta^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial\theta^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial\theta^{\prime}}\right)^{2}<br />
<br /> \frac{\partial x}{\partial\theta^{\prime}}=\frac{\partial x}{\partial\theta}=r\cos\theta\cos\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\cos\theta^{\prime}\cos\phi^{\prime}<br />
<br /> \frac{\partial y}{\partial\theta^{\prime}}=\frac{\partial y}{\partial\theta}=r\cos\theta\sin\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\cos\theta^{\prime}\sin\phi^{\prime}<br />
<br /> \frac{\partial z}{\partial\theta^{\prime}}=\frac{\partial z}{\partial\theta}=-r\sin\theta=-\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}<br />
<br /> Q_{\theta^{\prime}}^{2}=\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}<br />
<br /> Q_{\phi^{\prime}}^{2}=\left(\frac{\partial x}{\partial\phi^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial\phi^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial\phi^{\prime}}\right)^{2}<br />
<br /> \frac{\partial x}{\partial\phi^{\prime}}=\frac{\partial x}{\partial\phi}=-r\sin\theta\sin\phi=-\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}\sin\phi^{\prime}<br />
<br /> \frac{\partial y}{\partial\phi^{\prime}}=\frac{\partial y}{\partial\phi}=r\sin\theta\cos\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}\cos\phi^{\prime}<br />
<br /> \frac{\partial z}{\partial\phi^{\prime}}=\frac{\partial z}{\partial\phi}=0<br />
<br /> Q_{\phi^{\prime}}^{2}=\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}\sin^{2}\theta^{\prime}<br />
Finally:
<br /> \epsilon_{r^{\prime}}=\epsilon\frac{\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}\sin\theta^{\prime}}{\frac{R_{2}}{R_{2}-R_{1}}}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)^{2}\sin\theta^{\prime}<br />
<br /> \epsilon_{\theta^{\prime}}=\epsilon\frac{\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}}{\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\sin\theta^{\prime}<br />
<br /> \epsilon_{\phi^{\prime}}=\epsilon\frac{\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}\left(r^{\prime}-R_{1}\right)}{\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\csc\theta^{\prime}<br />
Any kind of help is more than welcome!
http://www.sciencemag.org/content/312/5781/1780.full
(can I post a link to this article in Dropbox so that people reading this don't have to register without getting an infraction from the moderators??)
and I'm getting this:
<br /> \epsilon'_{r'} = \epsilon \frac{R_2}{R_2-R_1} (r'-R_1)^2 \sin(\theta')<br />
<br /> \epsilon'_{\theta'} = \epsilon \frac{R_2}{R_2-R_1} \sin(\theta')<br />
<br /> \epsilon'_{\phi'} = \epsilon \frac{R_2}{R_2-R_1} \sin(\theta')<br />
instead of equations (7) in Pendry's article:
<br /> \epsilon'_{r'} = \frac{R_2}{R_2-R_1} \frac{(r'-R_1)^2}{r'}<br />
<br /> \epsilon'_{\theta'} = \frac{R_2}{R_2-R_1}<br />
<br /> \epsilon'_{\phi'} = \frac{R_2}{R_2-R_1}<br />
The difference between these equations and the ones I get is not only the missing r' and the extra sin(\theta') but also the extra \epsilon
This is what I'm doing. The new coordinates are given by equations (6):
<br /> r^{\prime}=R_{1}+r\frac{\left(R_{2}-R_{1}\right)}{R_{2}}<br />
<br /> \theta^{\prime}=\theta<br />
<br /> \phi^{\prime}=\phi<br />
The permittivity transforms according to:
<br /> \epsilon_{r}^{\prime}=\epsilon\frac{Q_{\theta'}Q_{\phi'}}{Q_{r'}}<br />
<br /> \epsilon_{\theta}^{\prime}=\epsilon\frac{Q_{r'}Q_{\phi'}}{Q_{\theta'}}<br />
<br /> \epsilon_{\phi}^{\prime}=\epsilon\frac{Q_{r'}Q_{\theta'}}{Q_{\phi'}}<br />
where Q_{u} is given by:
<br /> Q_u^2 = \left(\frac{\partial x}{\partial u}\right)^2 + \left(\frac{\partial y}{\partial u}\right)^2 + \left(\frac{\partial z}{\partial u}\right)^2<br />
Then:
<br /> Q_{r^{\prime}}^{2}=\left(\frac{\partial x}{\partial r^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial r^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial r^{\prime}}\right)^{2}<br />
<br /> \frac{\partial x}{\partial r^{\prime}}=\frac{\partial x}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial x}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial x}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial x}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\sin\theta\cos\phi\frac{R_{2}}{R_{2}-R_{1}}=\sin\theta^{\prime}\cos\phi^{\prime}\frac{R_{2}}{R_{2}-R_{1}}<br />
<br /> \frac{\partial y}{\partial r^{\prime}}=\frac{\partial y}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial y}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial y}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial y}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\sin\theta\sin\phi\frac{R_{2}}{R_{2}-R_{1}}=\sin\theta^{\prime}\sin\phi^{\prime}\frac{R_{2}}{R_{2}-R_{1}}<br />
<br /> \frac{\partial z}{\partial r^{\prime}}=\frac{\partial z}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial z}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial z}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial z}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\cos\theta\frac{R_{2}}{R_{2}-R_{1}}=\cos\theta^{\prime}\frac{R_{2}}{R_{2}-R_{1}}<br />
<br /> Q_{r^{\prime}}^{2}=\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}<br />
<br /> Q_{\theta^{\prime}}^{2}=\left(\frac{\partial x}{\partial\theta^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial\theta^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial\theta^{\prime}}\right)^{2}<br />
<br /> \frac{\partial x}{\partial\theta^{\prime}}=\frac{\partial x}{\partial\theta}=r\cos\theta\cos\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\cos\theta^{\prime}\cos\phi^{\prime}<br />
<br /> \frac{\partial y}{\partial\theta^{\prime}}=\frac{\partial y}{\partial\theta}=r\cos\theta\sin\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\cos\theta^{\prime}\sin\phi^{\prime}<br />
<br /> \frac{\partial z}{\partial\theta^{\prime}}=\frac{\partial z}{\partial\theta}=-r\sin\theta=-\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}<br />
<br /> Q_{\theta^{\prime}}^{2}=\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}<br />
<br /> Q_{\phi^{\prime}}^{2}=\left(\frac{\partial x}{\partial\phi^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial\phi^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial\phi^{\prime}}\right)^{2}<br />
<br /> \frac{\partial x}{\partial\phi^{\prime}}=\frac{\partial x}{\partial\phi}=-r\sin\theta\sin\phi=-\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}\sin\phi^{\prime}<br />
<br /> \frac{\partial y}{\partial\phi^{\prime}}=\frac{\partial y}{\partial\phi}=r\sin\theta\cos\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}\cos\phi^{\prime}<br />
<br /> \frac{\partial z}{\partial\phi^{\prime}}=\frac{\partial z}{\partial\phi}=0<br />
<br /> Q_{\phi^{\prime}}^{2}=\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}\sin^{2}\theta^{\prime}<br />
Finally:
<br /> \epsilon_{r^{\prime}}=\epsilon\frac{\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}\sin\theta^{\prime}}{\frac{R_{2}}{R_{2}-R_{1}}}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)^{2}\sin\theta^{\prime}<br />
<br /> \epsilon_{\theta^{\prime}}=\epsilon\frac{\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}}{\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\sin\theta^{\prime}<br />
<br /> \epsilon_{\phi^{\prime}}=\epsilon\frac{\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}\left(r^{\prime}-R_{1}\right)}{\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\csc\theta^{\prime}<br />
Any kind of help is more than welcome!