Show all your work for each part of the question. The parts within the question may not have equal weight.
A nonconducting sphere of radius \( R \) is centered on the origin. The sphere has a negative uniform volume charge density \( \rho_{\text {sphere }}=-\rho_{0} \). The sphere is surrounded by a nonconducting spherical shell of inner radius \( R \) and outer radius \( 2 R \). The shell has a positive uniform volume charge density \( \rho_{\text {shell }}=+\rho_{0} \), as shown in the figure. Points \( \mathrm{A}, \mathrm{B} \), and C are located on the \( x \)-axis at positions \( x=0, x=2 R \), and \( x=3 R \), respectively.
The magnitude of the electric fields at points \( \mathrm{A}, \mathrm{B} \), and C are \( E_{\mathrm{A}}, E_{\mathrm{B}} \), and \( E_{\mathrm{C}} \) respectively.
(b) Derive an expression for \( E_{\mathrm{B}} \) in terms of \( R, \rho_{0} \), and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference booklet.
