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Homework Statement
A) Use Gauss's Law to derive the electric field in all space for a nonconducting sphere with volumetric charge distribution ρ=ρ_{0}r^{3} and radius, R.
B) Repeat when there is a concentric spherical cavity within the non conducting sphere with radius, A.
Homework Equations
∫E⋅dA=Q_{enclosed}/ε_{0}
ρ=Q/V (charge density)
The Attempt at a Solution
Im struggling with part B more than A, but since B uses part A's answer (in a way) I will show both my attempts
A) I drew a picture (sphere.jpg attached) and will use words to explain what Im doing.
My sphere has radius, R.
My Gaussian Surface has radius, A.
I will be finding dQ of a small spherical piece of the sphere, the radius of this small spherical piece is, r, and has thickness, dr.
First find dQ
we know that ρ=Q/V →dQ=ρdV since ρ=ρ_{0}r^{3} and dV=(4πr^{2})(dr) lets plug those into our dQ:
dQ=(ρ_{0}r^{3})(4πr^{2})(dr)
We can now find our Q_{enclosed} of gaussian surface.
My below integral's limits go from 0→a (a being the radius of my gaussian surface)
Q_{enclosed}=∫dQ=∫ (ρ_{0}r^{3})(4πr^{2})(dr)=4πρ_{0}∫r^{4}→once evaluated= (4πρ_{0}a^{5})/5 (this is our Q_{enclosed})
Now we can use gauss's law to find electric field
E(4πa^{2})=(4πρ_{0}a^{5})/(5ε_{0})
and finally:
E=(ρa^{3})/(5ε_{0})→E(r)=ρr^{3})/(5ε_{0})r(hat)
This is the answer if you are inside the sphere at some distance r<R
Now for outside the sphere:
We can go to the Q_{enclosed} and just change the limits, instead of integrating from 0→a, we can integrate from 0→R. If we do this our Q_{enclosed} becomes:
Q_{enclosed}=(4πρ_{0}R^{5})/5
My new gaussian surface has radius a'
E(4πa'^{2})=(4πρ_{0}R^{5})/(5ε_{0})
solving for E:
E=(ρ_{0}R^{5})/(5a'^{2}ε_{0})
Now for part B
Again, new limits for Q_{enclosed}, the limits now are going from A→a
our new Q_{enclosed}=[(4πρ_{0}a^{5})/(5)][(4πρ_{0}A^{5})/(5)]
and our new E=[(ρ_{0}a^{3})/(5ε_{0})][(ρ_{0}A^{5})/(5a^{2}ε_{0})
and outside our Q_{enclosed}=[(4πρ_{0}R^{5})/(5)][(4πρ_{0}A^{5})/(5)]
and our new E=[(ρ_{0}R^{5})/(5a^{2}ε_{0})][(ρ_{0}A^{5})/(5a^{2}ε_{0})
Hope that was clear enough, please ask for more steps if Im wrong.
thanks!
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