Electric Potential Energy Spherical Shells

  • Thread starter JosephK
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  • #1
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Homework Statement



A solid sphere of radius R has a uniform charge density ρ and total charge Q. Derive an expression for its total electric potential energy. Suggestion: Imagine that the sphere is constructed by adding successive layers of concentric shells of charge dq = (4[itex]\pi[/itex] r[itex]^{2}[/itex] dr) ρ and let dU = Vdq. (Use any variable or symbol stated above along with the following as necessary: ke.)

Homework Equations



U = [itex]\int[/itex]4[itex]\pi[/itex]r[itex]^{2}[/itex]k[itex]_{e}[/itex][itex]\frac{q}{r}[/itex]dr

[itex]\rho[/itex]=[itex]\frac{Q}{\frac{4}{3}\pi r^{3}}[/itex]



The Attempt at a Solution



The sum of all dq is Q.

U = qV - q is test charge
U = q k[itex]_{e}[/itex][itex]\frac{Q}{r}[/itex] - equation of voltage substituted

dQ = dq k[itex]_{e}[/itex][itex]\frac{Q}{r}[/itex] -small potential energy with respect to small charge

dQ = 4[itex]k_{e}\pi\rho\frac{Q}{r} r^2 dr[/itex] - dq plugged in

Then I integrated both sides.
 

Answers and Replies

  • #2
Delphi51
Homework Helper
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I'm having a little trouble following that.
It seems to me the dQ for the spherical shell is 4πR²ρ*dR.
The work done to bring dQ in from infinity to R is dU = kQ/R*dQ.
And Q up to radius R is 4/3*πR³ρ.
Combined, dU = 16/3π²k ρ²R⁴dR
Check carefully; I make mistakes.
 
  • #3
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Thank you
 

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