Electric flux leaving a sphere

[ViolentCorpse]

Homework Statement

Within the spherical shell, 3 < r < 4 m, the electric flux density is given as D = 5(r − 3)3 ar C/m2
a) What is the volume charge density at r = 4?
b) How much electric flux leaves the sphere r = 4?

Homework Equations

ρ_v=Div D
Electric flux = ∫_sD.ds=∫_vρ_vdv

The Attempt at a Solution

I got the correct answer for part a which is 17.5 C/m³. My confusion is in part b. I'm only getting the correct answer by using Electric flux = ∫_sD.ds. Multiplying volume charge density by the volume of the sphere gives me the wrong answer and I don't understand why.

Please help me figure out my mistake.

Thank you so much!

 

[hjelmgart]

Why is question b formulated as if it's a sphere of radius 4? Is that correct, or do you still use the volume of a spherical shell?

 

[ViolentCorpse]

I'm sorry but I didn't get you..

 

[hjelmgart]

Within the spherical shell, 3 < r < 4 m, the electric flux density is given as D = 5(r − 3)3 ar C/m2
a) What is the volume charge density at r = 4?

Here you have a spherical shell

b) How much electric flux leaves the sphere r = 4?

Here it is a sphere of radius 4

So are they just using sphere to shorten it, or is it an entirely new geometrical object? That is what I was wondering. I was just puzzled by that formulation.

But as I can see, you are using the correct formula, I don't know why those 2 don't give the same result. I didn't try to do the actual calculations though.

 

[ViolentCorpse]

Oh, I see. I used the volume of the sphere i.e (4/3)*pi*r³

What changes should I make if I assume they're talking about a spherical shell and want to find the volume of a spherical shell?

Thanks

 

[hjelmgart]

You take the volume of the outer sphere minus that of the inner, which gives you that of the shell.

So you still have the same surface area, which should explain why your formulas for the electric flux don't add up.

 

[ViolentCorpse]

Okay I tried evaluating the volume integral from r=3 to r=4 and the answers still don't match.

Anyway, I appreciate your help, hjelmgart. Thanks a lot! :)