Two spheres, each of radius R and carrying uniform volume

QUESTION:

Problem 18P

Two spheres, each of radius R and carrying uniform volume charge densities +ρ and −ρ, respectively, are placed so that they partially overlap (Fig. 2.28). Call the vector from the positive center to the negative center d. Show

that the field in the region of overlap is constant, and find its value. [Hint: Use the answer to Prob. 2.12.]

Reference: Fig. 2.28.

Reference: Prob. 2.12.

Use Gauss’s law to find the electric field inside a uniformly charged solid sphere (charge density ρ). Compare your answer to Prob. 2.8.

Reference: Prob.2.8.

Use your result in Prob. 2.7 to find the field inside and outside a solid sphere of radius R that carries a uniform volume charge density ρ. Express your answers in terms of the total charge of the sphere, q. Draw a graph of |E| as a function of the distance from the center.

Reference: Prob.2.7

Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density σ. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write r in terms of R and θ. Be sure to take the positive square root:  if R > z, but it’s (z − R) if R < z.]

Reference: Fig.2.11.