Finding the Electric Field Outside of a Spherical Shell Using Gauss's Law

[skate_nerd]

Homework Statement

The problem states that you have a spherical shell with inner radius R_i=1 cm and outer radius R₀=2 cm. The shell also has uniform charge density of ρ=10^-3 N/m³. I found the first few answers of the question already. First was to get the charge of the shell, which is simply ρV_shell, or Q=ρ(4∏/3)(R₀³-R_i³). This ends up being 2.93(10^-8)C.
Next I found that the electric field magnitude everywhere inside the shell could be expressed in terms of r by just setting EA=ρ(4∏/3)(r³-R_i³)/ε₀, where the 4∏ cancels. the answer ends up being ((1/3)ρ/ε₀)(r-(R_i³/r²), or 3.76(10⁷) N/Cm.
Now the part I'm having trouble with is finding an equation in terms of r for the electric field magnitude outside of the spherical shell.

Homework Equations

Gauss's Law. Surface area and volume of a sphere.

The Attempt at a Solution

I tried using Gauss's law in a similar way to the last part of the problem by having EA=Q/ε₀, but I don't understand how you are supposed to solve for this if you don't know what the uniform charge density is outside of the shell. I don't feel like using that same value would make sense. Could somebody explain this to me?

 

[haruspex]

Next I found that the electric field magnitude everywhere inside the shell could be expressed in terms of r by just setting EA=ρ(4∏/3)(r³-R_i³)/ε₀, where the 4∏ cancels. the answer ends up being ((1/3)ρ/ε₀)(r-(R_i³/r²), or 3.76(10⁷) N/Cm.

How did you get that? There should not be a field inside an empty uniformly charged spherical shell.

 

[skate_nerd]

I said that is the E field inside the shell, as in like in the volume of the shell. Thats why I used the gaussian radius as arbitrary r minus the inner radius of the sphere.

 

[haruspex]

I said that is the E field inside the shell, as in like in the volume of the shell. Thats why I used the gaussian radius as arbitrary r minus the inner radius of the sphere.

OK - that's often a tricky distinction to make verbally.
The field outside a uniformly charged spherical shell happens to be exactly as though all the charge were concentrated at the sphere's centre.

I don't feel like using that same value would make sense.

Can you explain that some more? What same value?