Why Do Solid and Hollow Spheres Show Different Electric Fields Inside?

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SUMMARY

The discussion centers on the differences in electric fields inside solid and hollow spheres as explained by Gauss's Law. In the first example, an insulating solid sphere with uniform volume charge density produces an electric field inside given by E = kQr/a³, while the hollow spherical shell in the second example results in an electric field of zero inside. The key distinction lies in the distribution of charge; the solid sphere has charge throughout its volume, whereas the hollow shell has charge only on its surface, leading to different electric field behaviors.

PREREQUISITES
  • Understanding of Gauss's Law
  • Familiarity with electric fields and charge distributions
  • Knowledge of insulating materials and their properties
  • Basic calculus for integrating charge distributions
NEXT STEPS
  • Study the derivation and applications of Gauss's Law in electrostatics
  • Explore the concept of electric fields in different charge distributions
  • Learn about the properties of insulating materials versus conductive materials
  • Investigate the mathematical integration of charge distributions in spherical coordinates
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Students of physics, particularly those studying electromagnetism, educators explaining electric fields, and anyone seeking to deepen their understanding of charge distributions and their effects on electric fields.

Acuben
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As I was studying gauss's law to understand it's concept (which I still do not understand)
I came across two similar (or same?) looking problems that seems to give different result

Homework Statement


example 1)
An insulating solid sphere of radius a has a uniform volume charge density (rho) and carries a total positive charge Q
a) find magnitude of E at a point outside the sphere
b) find magnitude of E at a point inside the sphere
answer a) E=kQ/r^2
answer b)kQr/a^3
note: textbook used Qin=(rho)*(V')
and EA=Qin/e
where e is permititvity of vacuum or something >_>
r is for radius of the gaussian sphere

example 2)
A thin spherical shell of radius a has a total charge Q distributed uniformly over its surface. Find Electric field at points
a) outside
b) inside
answer a) E=kQ/r^2
answer b) E=0
note: not much work shown here

These are examples problems meaning all the work is shown on the textbook
example 1a sounds logical since E=kQ/r^2 is the equation of Electric field.
Even it is on gaussian sphere, it wouldn't make a difference which is same for example 2a.
Now for example 1b and 2b, why is it that two examples that seems to be the same problems
(both involving sphere). gives two different E (while they are both spheres of radius r and charge Q
distributed uniformly)It did say say example 1 is insulating solid sphere. I'm thinking maybe this is the reason.
In insulating sphere where charge is spread out, there are charges inside the shell, not just on the outside. (although what I still do not understand is that solid sphere should still have E=0 since hollow spherical shell have E=0, and solid sphere can be viewed as sum of thin hollow spherical shell (if you consider thin hollow spherical shell as dA but Integral of dA would be 0 since dA is 0... >_> I think I'm wrong but I do not know why.)
and example 2 is thin spherical shell. Does this make a difference?
or do I count first one as filled in sphere and 2nd one as hollow sphere?

Homework Equations


The Attempt at a Solution

 
Last edited:
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Acuben said:
In insulating sphere where charge is spread out, there are charges inside the shell, not just on the outside. (although what I still do not understand is that solid sphere should still have E=0 since hollow spherical shell have E=0, and solid sphere can be viewed as sum of thin hollow spherical shell (if you consider thin hollow spherical shell as dA but Integral of dA would be 0 since dA is 0... >_> I think I'm wrong but I do not know why.)
The field inside a hollow spherical shell is zero, but not outside, as the first example shows. If you think of a solid sphere as an infinite series of shells (which is totally fine), the shells outside would contribute nothing to the field (as you say); but the shells farther inside would contribute to the field at a given point.

You should work on understanding gauss' law. Is there something in particular that doesn't make sense?
 
that makes sense, thank you =>
 

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