- #1

- 307

- 0

## Homework Statement

We know that a sphere of radius R carries a polarization

**P**(

**r**)=k

**r**where k is a constant and

**r**the vector from the centre. Calculate [tex]\sigma_b[/tex] and [tex]\rho_b[/tex]

## The Attempt at a Solution

If we let the direction of polarization coincide with the z axis then:

[tex]\sigma_b=\textbf{P}\hat{n}[/tex]

[tex]\sigma_b=kr\cos\theta[/tex]

and

[tex]\rho_b=-\nabla\textbf{P}[/tex]

[tex]\rho_b=-\nabla kr\hat{r}[/tex]

[tex]\rho_b=-kr (\hat{x}\frac{d}{dx}+\hat{y}\frac{d}{dy}+\hat{z}\frac{d}{dz}) (\sin\theta\cos\phi\hat{x}+\sin\theta\sin\phi\hat{y}+\cos\theta\hat{z})[/tex]

[tex]\rho_b=0[/tex]

My query is mostly about these last three steps...are they correct? I'm still struggling to wrap my head around substituting the right infinitesimals for integration problems and, as in this case, the right coordinates for vectors.

Thanks for the help!

phyz