PhDeezNutz
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- Homework Statement
- In the process of trying to derive an expression for the vector potential of a radiating magnetic dipole(/electric quadrupole)
I came up with this term
$$\vec{I} = \int \left(\hat{r}\cdot\vec{r'}\right) \vec{r'} \rho\left( \vec{r'} \right)\, d^3r'$$
and I want to massage it into a form that looks like a tensor product of sorts.
- Relevant Equations
- The non-traceless quadrupole is defined as
$$Q_{ij} = \int r_i r_j \rho\left( \vec{r'}\right) \, d^3r'$$
Before writing out each component I'm going to simplify ##\vec{I}## to the best of my abilities
$$\vec{I} = \int \left(\hat{r}\cdot\vec{r'}\right) \vec{r'} \rho\left( \vec{r'} \right)\, d^3r'$$
$$\vec{I} = \hat{r} \cdot \int \vec{r'} \left( x' , y', z' \right) \rho\left( \vec{r'} \right)\, d^3r'$$
Now I'm going to concentrate on each component
$$I_x =\hat{r} \cdot \int x' \vec{r}' \rho \left( \vec{r'} \right) \, d^3r'$$
$$I_y =\hat{r} \cdot \int y' \vec{r}' \rho \left( \vec{r'} \right) \, d^3r'$$
$$I_z=\hat{r} \cdot \int z' \vec{r}' \rho \left( \vec{r'} \right) \, d^3r'$$
Written out completely that is
$$I_x = \left( \hat{r} \cdot \int x'^2 \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{x} + \left(\hat{r} \cdot \int x'y' \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{y} + \left(\hat{r} \cdot \int x'z' \rho \left( \vec{r'} \right)\, d^3r'\right) \hat{z}$$
$$I_y = \left( \hat{r} \cdot \int y'x' \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{x} + \left(\hat{r} \cdot \int y'^2 \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{y} + \left(\hat{r} \cdot \int y'z' \rho \left( \vec{r'} \right)\, d^3r'\right) \hat{z}$$
$$I_z = \left( \hat{r} \cdot \int z'x' \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{x} + \left(\hat{r} \cdot \int z'y' \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{y} + \left(\hat{r} \cdot \int z'^2 \rho \left( \vec{r'} \right)\, d^3r'\right) \hat{z}$$
I'm trying to get this in a compact form that involves the quadrupole tensor but I can't seem to do it.
As always any help is appreciated in advanced.
$$\vec{I} = \int \left(\hat{r}\cdot\vec{r'}\right) \vec{r'} \rho\left( \vec{r'} \right)\, d^3r'$$
$$\vec{I} = \hat{r} \cdot \int \vec{r'} \left( x' , y', z' \right) \rho\left( \vec{r'} \right)\, d^3r'$$
Now I'm going to concentrate on each component
$$I_x =\hat{r} \cdot \int x' \vec{r}' \rho \left( \vec{r'} \right) \, d^3r'$$
$$I_y =\hat{r} \cdot \int y' \vec{r}' \rho \left( \vec{r'} \right) \, d^3r'$$
$$I_z=\hat{r} \cdot \int z' \vec{r}' \rho \left( \vec{r'} \right) \, d^3r'$$
Written out completely that is
$$I_x = \left( \hat{r} \cdot \int x'^2 \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{x} + \left(\hat{r} \cdot \int x'y' \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{y} + \left(\hat{r} \cdot \int x'z' \rho \left( \vec{r'} \right)\, d^3r'\right) \hat{z}$$
$$I_y = \left( \hat{r} \cdot \int y'x' \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{x} + \left(\hat{r} \cdot \int y'^2 \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{y} + \left(\hat{r} \cdot \int y'z' \rho \left( \vec{r'} \right)\, d^3r'\right) \hat{z}$$
$$I_z = \left( \hat{r} \cdot \int z'x' \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{x} + \left(\hat{r} \cdot \int z'y' \rho \left( \vec{r'} \right)\, d^3r' \right) \hat{y} + \left(\hat{r} \cdot \int z'^2 \rho \left( \vec{r'} \right)\, d^3r'\right) \hat{z}$$
I'm trying to get this in a compact form that involves the quadrupole tensor but I can't seem to do it.
As always any help is appreciated in advanced.
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