# Problem: Cartesian tensor to Spherical Tensor

gaffareee
Dear Frnds,

I have a dielectric tensor like:

epsilon_T=
|1 0 0|
|0 1 0| %% note that it is in cartesian co-ordinate system
|0 0 a|

now the potential equation of dipole should be as in spherical system

V=1/(4*pi*epsilon_T (?) *epsilon_0)*(qdcos(theta)/R)

now should i change the epsilon_T value cartesian to spherical system ?

or how i can write down the code ?

should it be V_xx, V_yy and V_zz?

## Answers and Replies

Homework Helper
Gold Member
now the potential equation of dipole should be as in spherical system

V=1/(4*pi*epsilon_T (?) *epsilon_0)*(qdcos(theta)/R)

This equation doesn't make much sense. On the LHS you have a scalar, while on the RHS you seem to have an uncontracted tensor in the denominator.

I believe that you should have an expression for the potential (electric or energy, it doesn't matter) that is proportional to

$$\sum_{ij} (\epsilon_T)_{ij} E_i r_j.$$

As a scalar, this quantity can be evaluated in Cartesian coordinates and then written in terms of spherical coordinates, which would avoid having to convert vectors to spherical coordinates.

Phrak
You haven't said what type tensor it is.

kanato
First write down the D field. For example, for a single point charge the D field is
$$\vec{D} = q/(4\pi r^2) \hat{r}$$

Then use the relationship $$\vec{D} = \epsilon \vec{E}$$ to get the E field (you have to invert your dielectric tensor).

Then perform the integral to calculate the potential from the E field. The potential will still be a scalar but I think you will find that instead of 1/r you will have something like 1/sqrt(x^2+y^2+(az)^2). But you have to check.