Question

You have learned to apply Gauss's law to spherical charged bodies with uniform charge density ρ. Suppose some tech genius approaches you and asks you to analyze a charged component that his company has developed and will feature in his next upcoming invention. This charged body that you need to study has a spherical shape of radius R and it contains positive charge Q. The tricky part is that its volume charge density is not constant but is a function of the distance from the center, i.e. ρ(r). This charge density is equal to 3Ar/(2R) for r ≤ R/2 and A[1 - (r/R)^2] for R/2 ≤ r ≤ R. The quantity A is a constant having units of C/m^3. 1. How would this constant A be expressed in terms of the total charge Q and radius R of the sphere? 2. Apply Gauss's law and determine the magnitude of the electric field as a function of r both inside and outside the sphere. (Inside the sphere, consider regions of two different volume charge densities separately.) 3. What fraction of the total charge is deposited in both regions of the sphere? 4. What is the magnitude of the electric field at point R/2?

          You have learned to apply Gauss's law to spherical charged bodies with uniform charge density ρ. Suppose some tech genius approaches you and asks you to analyze a charged component that his company has developed and will feature in his next upcoming invention. This charged body that you need to study has a spherical shape of radius R and it contains positive charge Q. The tricky part is that its volume charge density is not constant but is a function of the distance from the center, i.e. ρ(r). This charge density is equal to 3Ar/(2R) for r ≤ R/2 and A[1 - (r/R)^2] for R/2 ≤ r ≤ R. The quantity A is a constant having units of C/m^3. 

1. How would this constant A be expressed in terms of the total charge Q and radius R of the sphere? 
2. Apply Gauss's law and determine the magnitude of the electric field as a function of r both inside and outside the sphere. (Inside the sphere, consider regions of two different volume charge densities separately.) 
3. What fraction of the total charge is deposited in both regions of the sphere? 
4. What is the magnitude of the electric field at point R/2?

Added by Jonathan G.

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