Conceptual question with Concentric Shell Potental

[Manodesi524]

Homework Statement

Homework Equations

The Attempt at a Solution

I'm able to arrive at the correct solution here, but I'm fairly sure that my line of reasoning is not proper.

1. For the outermost shell, I discard the inner two shell/charge, since their effective charge is zero. Is this a correct assumption?
2. In computing the potential between R1 and R2, I discard the space between the point charge and spherical shell since there is no net electric field emanating from them. This gives a potential of 0 for that middle Then, I add on the contribution from the outmost shell, which was previously computed. Is adding on the electric potential of the outermost shell without respect to the innermost shell appropriate here at the end?
3. Here is where I get very confused. I start by taking the potential from the point charge to R1. Simple enough. However, we previously calculated the potential from R1 to infinity in step 2, so my instinct would be to simply add that on since potential is a scalar. However, the final answer has a contribution from the middle component (-2Qk/R1) in it. Why does the middle layer suddenly start contributing?

 

[kuruman]

1. For the outermost shell, I discard the inner two shell/charge, since their effective charge is zero. Is this a correct assumption?

It's not an assumption. If you use Gauss's law in the region r > R₂, the enclosed charge is +Q as if the inner charges weren't there.

This gives a potential of 0 for that middle

No. Again, application of Gauss's law says that the electric field is zero because the enclosed charge is zero. This makes the potential constant but not necessarily zero in that region.

3. Here is where I get very confused.

You will avoid confusion if
(a) You use Gauss's law first to find the electric field in all three regions.
(b) Integrate, starting at infinity. You need to do three integrals, one for each region using the expression for the E-field that you found in (a).