1- Suppose that there is a dielectric sphere (a sphere made up of a medium of dielectric constant e) of radius a centered at the origin. Suppose also that a point charge q is located in free-space on the z-axis at a distance b (a < b) from the origin a) Determine the electrostatic potential V at all points in space (i.e. in the three separate regions ra, arband b < r) using Legendre polynomials. Hint: The solution for each region is given in terms of infinite series of the Legendre polynomials Pi(cos 0), l E N. The method of images doesn't work here, at least with a simple setting as for the semi-infinite slab of dielectric case b) Verify that in the limit e -o0, your answer to part a) reduces to the solution for the analogous problem with the conducting sphere

1- Suppose that there is a dielectric sphere (a sphere made up of a medium of dielectric constant e) of radius a centered at the origin. Suppose also that a point charge q is located in free-space on the z-axis at a distance b (a < b) from the origin a) Determine the electrostatic potential V at all points in space (i.e. in the three separate regions ra, arband b < r) using Legendre polynomials. Hint: The solution for each region is given in terms of infinite series of the Legendre polynomials Pi(cos 0), l E N. The method of images doesn't work here, at least with a simple setting as for the semi-infinite slab of dielectric case b) Verify that in the limit e -o0, your answer to part a) reduces to the solution for the analogous problem with the conducting sphere