A sphere of radius R is made of a dielectric material with a permanent \"frozen-in\" polarization polarization that varies with position and points outward in the radial direction: \(\vec{P} = cr^2\hat{r}\), where c is a constant, r is the distance from the center of the sphere, and \(\hat{r}\) is the unit vector in the radial direction.
(a) Using the formulas \(\rho_b = -\nabla \cdot \vec{P}\) and \(\sigma_b = \vec{P} \cdot \hat{n}\) calculate the bound polarization charge density in the volume of the sphere and at the surface of the sphere. Remember that the normal unit vector points out of the dielectric. [2.5 points]
(b) Integrate your results from part (a) to find the total bound charge inside the sphere and at its surface. [2.5 points]
(c) Using Gauss's law, calculate the electric field inside the sphere (r < R) and outside the sphere (r > R). [2.5 points]
(d) Using the formula \(\vec{D} = \epsilon_0 \vec{E} + \vec{P}\), calculate the electric displacement \(\vec{D}\) both inside and outside the sphere. [2.5 points]
