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1. The cavity inside a neutral conducting spherical shell of inner radius a and outer radius b is filled with an insulating material that has a non-uniform charge density p(r) = Porla, where po is a positive constant. a. Find the electric potential as a function of r in every region of space taking V (r → ∞) = 0. Hint: Find the electric field first using Gauss' Law and start finding the potential in the region r> b to work your way inwards. b. Test your results from (a) by working your way back to obtain the known electric field using ar c. A point charge q> 0 with mass m is moving from infinitely far away towards the center of the sphere in the radial direction. What is the speed of this point charge at r= 2b if it were to barely touch the outer surface of the shell before being reflected?

1. The cavity inside a neutral conducting spherical shell of inner radius a and outer radius b is filled with an insulating material that has a non-uniform charge density p(r) = Porla, where po is a positive constant. a. Find the electric potential as a function of r in every region of space taking V (r → ∞) = 0. Hint: Find the electric field first using Gauss' Law and start finding the potential in the region r> b to work your way inwards. b. Test your results from (a) by working your way back to obtain the known electric field using ar c. A point charge q> 0 with mass m is moving from infinitely far away towards the center of the sphere in the radial direction. What is the speed of this point charge at r= 2b if it were to barely touch the outer surface of the shell before being reflected?

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1. The cavity inside a neutral conducting spherical shell of inner radius a and outer radius b is filled with an insulating material that has a non-uniform charge density p(r) = Porla, where po is a positive constant. a. Find the electric potential as a function of r in every region of space taking V (r→∞) = 0. Hint: Find the electric field first using Gauss' Law and start finding the potential in the region r> b to work your way inwards. b. Test your results from (a) by working your way back to obtain the known electric field using av ar c. A point charge q> 0 with mass m is moving from infinitely far away towards the center of the sphere in the radial direction. What is the speed of this point charge at r= 2b if it were to barely touch the outer surface of the shell before being reflected?
Transcribed Image Text:1. The cavity inside a neutral conducting spherical shell of inner radius a and outer radius b is filled with an insulating material that has a non-uniform charge density p(r) = Porla, where po is a positive constant. a. Find the electric potential as a function of r in every region of space taking V (r→∞) = 0. Hint: Find the electric field first using Gauss' Law and start finding the potential in the region r> b to work your way inwards. b. Test your results from (a) by working your way back to obtain the known electric field using av ar c. A point charge q> 0 with mass m is moving from infinitely far away towards the center of the sphere in the radial direction. What is the speed of this point charge at r= 2b if it were to barely touch the outer surface of the shell before being reflected?
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Given:

The non-uniform  charge density of the spherical shell is ρ(r) = ρ0 ra

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