Thick spherical shell confirmation of solution

[Liquidxlax]

Homework Statement

A spherical shell of inner radius a and outer radius b contains a uniform charge density ρ of total charge Q. (a) Find ρ (b) find the electric field for r< a, a< r< b, b< r sketch it

Homework Equations

E = kQ/r²

V=4πr³

The Attempt at a Solution

ρ π

since it is a thick shell the volume would be V = (4/3)π(b³ - a³)

and ρV = Q_enc => ρ=Q/V

so

ρ = 3Q/(4π(b³ - a³))


(b) r<a the electric field will be 0 because the electric field can't close on itself

a< r< b

E=kQ_enc/r² = (4πkρ(b³ - a³))/3r³


b<r kQ/r² because the sphere can be treated as a point charge and the electric field is symmetric on a sphere


Does this look right?

 

[collinsmark]

E= [...] = (4πkρ(b³ - a³))/3r³

There's two errors with the above equation (see the red).

 

[Liquidxlax]

There's two errors with the above equation (see the red).

okay i see how the r³ is wrong, that was just a transcribing error, but how come the b is wrong? or would it be because we know that r<b so we can assume it to be

(r³-a³) because a is constant and r does not exceed b

 

[collinsmark]

There you go! Remember Gauss' law relates to the charge enclosed within the Gaussian surface (the Gaussian surface in this case has radius r). In the case where b > r > a, then any charge between r and b is outside the Gaussian surface, and has no effect (in this spherically symmetrical case).