a. A fixed volume charge distribution of constant charge density p_0 is contained within a rectangular box centered at the origin of a Cartesian coordinate system (x, y, z). The box has dimensions w X w X d, where d << w ( i.e. you can model the sheet as infinite in the x and y directions).

i) Find the total charge inside the box

ii) Find the electric field E(z) on the z-axis above, below and inside the box under the assumption that z << w.

iii) On the z-axis, find the difference in the electric potential between the bottom and center of the box i.e., find V(z = 0) - V(z = -d/2), and between the bottom and the top of the box i.e., find V(z = d/2) -V(z = -d/2).

b. A large plane parallel slab of a linear homogeneous dielectric material also of thickness f with a relative permittivity, E_p is placed on the top surface of the charged box of question 2(a).

i) Find the electric field inside the diaelectric.

ii) Fine the polarization vector P on the z-axis

iii) Find the bound volume and surface polarization charge densities along the z- axis

c. Consider a charge distribution enclosed in a box with the same geometry as the box in question 2(a) and 2(b), but rather than having a constant charge density, the charge density varies inside the box with z as p(z) = k(z + d/2), where k is a constant.

i) Fine the total charge inside the box.

ii) Find the electric field E(z) on the z-axis inside the box.

iii) On the z-axis, find the difference in the electric potential between the bottom and center of the box, i.e., find V(z = 0) - V(z = -d/2), and between the bottom and the top of the box, i.e., find V(z = d/2) - V(z = -d/2).