Help with electrostatics problem (spherical shell charge distribution)

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SUMMARY

The forum discussion centers on a static electric charge distributed in a spherical shell with inner radius R1 and outer radius R2, where the charge density is defined as ρ=a+br. The professor critiques the solution provided in the referenced book, specifically on pages 20-21, asserting that the integral calculations for the electric field and potential are incorrect. The participants emphasize the importance of correctly interpreting the charge distribution and the implications of the electric field being factored out of the integrals, which leads to misunderstandings in the solution.

PREREQUISITES
  • Understanding of electrostatics principles, particularly electric fields and potentials.
  • Familiarity with charge density functions and their implications in spherical coordinates.
  • Proficiency in performing integrals related to electric fields and potentials.
  • Knowledge of the concept of electric potential being zero at infinity.
NEXT STEPS
  • Review the derivation of electric fields from charge distributions in spherical coordinates.
  • Study the implications of charge density variations on electric field calculations.
  • Learn about the mathematical techniques for integrating functions in electrostatics.
  • Examine the concept of electric potential and its relationship with electric fields in different regions.
USEFUL FOR

This discussion is beneficial for physics students, educators in electromagnetism, and anyone involved in solving electrostatics problems, particularly those related to spherical charge distributions.

sroot
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Homework Statement
A static electric charge is distributed in a spherical shell of inner radius R1 and outer radius R2. The electric charge density is given by ρ=a+br, where r is the distance from the centre, and zero everywhere else.
(i) Find an expression for the electric field everywhere in terms of r.
(ii) Find expressions for the electric potentials and energy density for r < R1. Take the potential to be zero at r→∞.
Relevant Equations
ρ=a+br
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sroot said:
Homework Statement:: A static electric charge is distributed in a spherical shell of inner radius R1 and outer radius R2. The electric charge density is given by ρ=a+br, where r is the distance from the centre, and zero everywhere else.
(i) Find an expression for the electric field everywhere in terms of r.
(ii) Find expressions for the electric potentials and energy density for r < R1. Take the potential to be zero at r→∞.
Relevant Equations:: ρ=a+br

According to my professor, the solution in this book (pages 20-21) for item (ii) is wrong: https://www.u-cursos.cl/usuario/754...roblems_and_Solutions_on_Electromagnetism.pdf
Welcome to PF.

Can you please just take screenshots of the pages that you want to discuss? Asking us to download the whole book PDF file is a bit too much. And please explain what is confusing you about the solution. Thank you.
 
berkeman said:
Welcome to PF.

Can you please just take screenshots of the pages that you want to discuss? Asking us to download the whole book PDF file is a bit too much. And please explain what is confusing you about the solution. Thank you.

This is the solution for item (b). According to the professor, the integral is completely off. I can't understand why.
Screenshot_2020-12-03 Problems_and_Solutions_on_Electromagnetism pdf.png
 
Did you carry out the integrations? What did you get for the results of the individual integrals?
 
It looks like the solution asserts that the electric field is the same in all regions by showing it factored out of the integrals. That is simply not true. I started doing the integrals but I quit before I got to the bottom line when it became obvious that my expression would not simplify to the answer shown above.

I interpreted "The electric charge density is given by ρ=a+br, where r is the distance from the centre, and zero everywhere else" to mean that there is charge only in the region ##R_1\leq r \leq R_2## "everywhere else" being outside this region. Otherwise why bother mentioning a shell?

Also, it is misleading (but not an error) to write the left hand side as ##\varphi (r)## when it is actually independent of ##r##.
 
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