I Appropriate coordinates for a given electric field

AI Thread Summary
The discussion focuses on the calculations of electric field divergence in different coordinate systems, specifically cylindrical and spherical coordinates. It emphasizes that the divergence value at a point should remain consistent across coordinate systems, meaning a zero divergence in one system should also yield zero in another. Participants express confusion over discrepancies in their calculations, particularly regarding the absence of a theta component in cylindrical coordinates and the resulting non-zero values in the radial direction. The importance of correctly applying the divergence formula for each coordinate system is highlighted, as it is not simply a matter of renaming variables from Cartesian coordinates. Overall, the conversation seeks clarity on the appropriate application of divergence in various coordinate systems.
kirito
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I was facing a problem where I have an electric field and I have to find the charge density , I noticed that the field outside is that of a dipole and because I decided to use spherical coordinates and reached an appropriate answer yet using divergence in cylindrical lead me to different outputs
Screenshot 2024-10-13 at 16.19.16.png

this is the field I was provided
and this is the charge density that I have reached
Screenshot 2024-10-13 at 16.21.52.png

Screenshot 2024-10-13 at 18.40.38.png

I tried to use this yet the output was different
I also used Cartesian it gave me the same output as the spherical ones
 
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The value of the divergence of a field at a point is independent of the coordinate system. So, if you get zero in spherical coordinates you should also get zero in cylindrical coordinates.
 
TSny said:
The value of the divergence of a field at a point is independent of the coordinate system. So, if you get zero in spherical coordinates you should also get zero in cylindrical coordinates.
thats actually why I even made this post , I got weirded out by the calculation using cylindrical coordinates since there is no component in the theta direction whereas the z results in 0 and the component in the r direction results in something other than zero unless I had to do something to the coordinates I changed from cylindrical to spherical so changed the z hat appropriately in the first case I may have missed something but got no clue what it may be
 
kirito said:
since there is no component in the theta direction whereas the z results in 0 and the component in the r direction results in something other than zero
What is the divergence in cylindrical (or spherical) coordinates? Hint: it is not the same as for Cartesian replacing variable names.
 
It's easier to do it without coordinates:
$${\bf E}=\frac{3{\bf (r\cdot \mu)r}}{r^5}-\frac{\mu}{r^3}$$
$$\nabla\cdot{\bf E}=\frac{3[\mu\cdot{\bf r}+3{\bf (\mu\cdot r)]}}{r^5}-\frac{15\mu\cdot{\bf r}}{r^5}+\frac{3\mu\cdot{\bf r}}{r^5}=0.$$
Sorry. I don't know what's wrong with my latex.
 
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