A point dipole 𝐩 is imbedded at the center of a sphere of...

A point dipole $\mathbf{p}$ is imbedded at the center of a sphere of linear dielectric material (with radius $R$ and dielectric constant $\epsilon_{r}$ ). Find the electric potential inside and outside the sphere $$\left[\text { Answer }: \frac{p \cos \theta}{4 \pi \epsilon r^{2}}\left(1+2 \frac{r^{3}}{R^{3}} \frac{\left(\epsilon_{r}-1\right)}{\left(\epsilon_{r}+2\right)}\right),(r \leq R) ; \quad \frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}}\left(\frac{3}{\epsilon_{r}+2}\right),(r \geq R)\right]$$

A point dipole $\mathbf{p}$ is imbedded at the center of a sphere of linear dielectric material (with radius $R$ and dielectric constant $\epsilon_{r}$ ). Find the electric potential inside and outside the sphere
$$\left[\text { Answer }: \frac{p \cos \theta}{4 \pi \epsilon r^{2}}\left(1+2 \frac{r^{3}}{R^{3}} \frac{\left(\epsilon_{r}-1\right)}{\left(\epsilon_{r}+2\right)}\right),(r \leq R) ; \quad \frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}}\left(\frac{3}{\epsilon_{r}+2}\right),(r \geq R)\right]$$

Key Concepts

Electric Dipole

An electric dipole is a pair of equal and opposite charges separated by a small distance, characterized by its dipole moment which quantifies the strength and orientation of the dipole. This concept is fundamental in electrostatics as it describes how localized charge distributions generate spatially varying electric fields, especially in regions distant relative to the separation scale of the charges.

Dielectric Materials

Dielectric materials are insulators that become polarized when subjected to an external electric field. The polarization of the medium is described by its dielectric constant (or relative permittivity), which quantifies the material’s ability to reduce the effective electric field within it. In problems involving dielectrics, the material response alters the distribution of the electric field and the potential compared to the vacuum case.

Electric Potential

Electric potential is a scalar quantity representing the potential energy per unit charge at a point in space due to an electric field. The potential is often determined by solving Laplace's or Poisson's equations and is particularly useful because electric field lines can be derived as the gradient of the potential, simplifying the analysis of electric field configurations.

Boundary Conditions in Electrostatics

Boundary conditions at interfaces between different media are essential for solving electric potential problems. At the interface between a dielectric material and its surroundings, the electric potential must be continuous, while the normal component of the electric displacement field is adjusted based on the difference in permittivity across the boundary. These conditions ensure the correct behavior of electric fields and potentials at material interfaces.

Multipole Expansion and Spherical Harmonics

Multipole expansion is a mathematical technique used to express potential fields due to localized charge distributions as a series of components (monopole, dipole, quadrupole, etc.). In problems with spherical symmetry or angular dependence, spherical harmonics provide a natural basis for expanding the potential. In the context of a dipole inside or outside a spherical region, the dipole term (l = 1) in the expansion is particularly relevant, and the associated Legendre polynomials facilitate the angular dependence characterization.

Please give Ace some feedback