Because there is a discontinuity in polarization at the...

Because there is a discontinuity in polarization at the boundary of the dielectric disc, a bound surface charge appears, which is the source of the electric field inside and outside the disc. We have for the electric field at the origin. $$ \vec{E}=-\int \frac{\sigma^{\prime} d S}{4 \pi \varepsilon_{0} r^{3}} \vec{r} $$ where $\vec{r}=$ radius vector to the origin from the element $d S$. $\sigma^{\prime}=P_{n}=P \cos \theta$ on the curved surface $\left(P_{n}=0\right.$ on the flat surface. $)$ Here $\theta=$ angle between $\vec{r}$ and $\vec{P}$ By symmetry, $\vec{E}$ will be parallel to $\vec{P}$. Thus $$ E=-\int_{0}^{2 \pi} \frac{P \cos \theta R d \theta \cdot \cos \theta}{4 \pi \varepsilon_{0} R^{2}} \cdot d $$ where, $\quad r=R$ if $d \ll R$. So, $E=-\frac{P d}{4 \varepsilon_{0} R}$ and $\vec{E}=-\frac{\vec{P} d}{4 \varepsilon_{0} R}$

Because there is a discontinuity in polarization at the boundary of the dielectric disc, a bound surface charge appears, which is the source of the electric field inside and outside the disc.
We have for the electric field at the origin.
$$
\vec{E}=-\int \frac{\sigma^{\prime} d S}{4 \pi \varepsilon_{0} r^{3}} \vec{r}
$$
where $\vec{r}=$ radius vector to the origin from the element $d S$.
$\sigma^{\prime}=P_{n}=P \cos \theta$ on the curved surface
$\left(P_{n}=0\right.$ on the flat surface. $)$
Here $\theta=$ angle between $\vec{r}$ and $\vec{P}$ By symmetry, $\vec{E}$ will be parallel to $\vec{P}$. Thus
$$
E=-\int_{0}^{2 \pi} \frac{P \cos \theta R d \theta \cdot \cos \theta}{4 \pi \varepsilon_{0} R^{2}} \cdot d
$$
where, $\quad r=R$ if $d \ll R$.
So, $E=-\frac{P d}{4 \varepsilon_{0} R}$ and $\vec{E}=-\frac{\vec{P} d}{4 \varepsilon_{0} R}$

Key Concepts

Symmetry in Electrostatics

The use of symmetry in electrostatic problems simplifies the calculation of electric fields. When the charge distribution displays symmetry, many vector components cancel out, reducing the complexity of the integration. This concept allows one to streamline the problem by focusing only on the components that contribute to the net field, ultimately leading to more straightforward analytical solutions.

Electric Field from Surface Charge Distributions

To calculate the electric field due to surface charges, one integrates the contributions from each infinitesimal surface element using Coulomb’s law. This involves summing the vector contributions of the electric field considering the inverse square dependence on the distance and directional properties relative to the field point. Such methods are fundamental in electrostatics and ensure that all effects due to surface charge distributions are accurately accounted for.

Bound Surface Charges

Bound surface charges arise at the interfaces of dielectric materials where the polarization changes discontinuously. These charges are not free to move throughout the material but are a result of the separation of positive and negative bound charges at the surface. Their presence is crucial in determining the electric field distribution both inside and outside the dielectric, as they act as sources in Coulomb’s law integration.

Polarization in Dielectric Materials

Polarization refers to the alignment of electric dipoles within a dielectric material when subjected to an electric field. This alignment creates a non-uniform distribution of microscopic charges that give rise to macroscopic effects. When there is a spatial discontinuity in polarization, such as at the boundary of a dielectric, the abrupt change leads to the appearance of bound charges that modify the electric field in the material and its surroundings.

Please give Ace some feedback