The energy needed to construct a charged hollow spherical shell with finite thickness

AI Thread Summary
The discussion focuses on calculating the energy required to construct a charged hollow spherical shell with inner radius r1 and outer radius r2, given a charge density p. The user attempts to divide the sphere into infinitesimal shells and derives the charge for each shell as 4πR^2*p*dr. They encounter a challenge with the dr^2 term in their integral and express concern that their method does not account for previously created shells. The conversation emphasizes the need to correctly integrate the work done in adding a thin shell of charge q to the existing thick shell. A proper approach involves considering the cumulative effect of all shells during the integration process.
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Homework Statement
I am trying to solve the relevant integral
Relevant Equations
I found that the energy needed to create a shell with radius R and charge Q is U=kQ^2/2R
So in the probleme is given a sphere with inner radius of r1 and outher radius r2 and elecric charge density p.

i tried to devide the sphere to shells with radius dr. And got that the charge for each shell is 4piR^2*p*dr
when plugin this expression to the work needed to create one shell i get dr^2.

so first of all how in general i can deal with dr^2 in integrals and secondly i guess that the way iam solving it is not taking into acount the shells created before so what is the right approach. Thanks.
 
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Consider the work done in adding a thin shell of charge q to an existing thick shell of charge density p.
 
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