# [Physics] Potential of surface charge

electromagnetismvectors

I have a question about the $\hat{n}$ in this formula $\sigma = P \dot{}\hat{n}$.

Why do sometime in my book they get $\sigma = P \cos{\theta}$ for a sphere. Isn't $\hat{n} = r$ ?

And then in another problem, where $P = \cfrac{k}{r}\hat{r}$ and r = a (inner) and r = b (outer) they get $\sigma =\cfrac{k}{a}$ and $\sigma =\cfrac{k}{b}$.

$\sigma$ = surface charge density

$P$ = polarization

In general, the surface charge density $\sigma$ of a polarized material with polarization $\mathbf{P}$ is indeed given by $\sigma=\mathbf{P}\cdot\hat{\mathbf{n}}$, where $\hat{\mathbf{n}}$ is the surface normal vector.
For a sphere, we have $\hat{\mathbf{n}}=\hat{\mathbf{r}}$ in spherical coordinates. So, assuming that $\mathbf{P}$ points in the $z$-axis, we obtain $$\sigma=\mathbf{P}\cdot\hat{\mathbf{r}}=P\hat{\mathbf{z}}\cdot\hat{\mathbf{r}}=P\cos{\theta}.$$
For the next case, we have a spherical shell with inner radius $a$ and outer radius $b$, with polarization $\mathbf{P}=\frac{k}{r}\hat{\mathbf{r}}$. Applying the general formula, we get
$$\sigma=\mathbf{P}\cdot\hat{\mathbf{r}}=\frac{k}{r}\hat{\mathbf{r}}\cdot\hat{\mathbf{r}}=\frac{k}{r}.$$ Letting $r=a$ and $r=b$ gives the surface charge density for the inner and outer surface respectively.