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## Homework Statement

Find the total electric charge in a spherical shell between radii

*a*and

*3a*when the charge density is:

ρ(r)=D(4a-r)

Where D is a constant and r is the modulus of the position vector r measured from the centre of the sphere

Where D is a constant and r is the modulus of the position vector r measured from the centre of the sphere

## Homework Equations

Q=ρV

Volume of a sphere = (4/3)πr

^{3}

## The Attempt at a Solution

My initial thinking was that I needed to get involved with the different forms of Gauss' Law, however the more I think about it the less I understand.

With the statement being a shell, should I consider two Gaussian surfaces at the various r values and sum the two charge values? Or should I do as I have done here and assume volumes as I have a charge density?

My attempt is:

When

*r=a*

ρ(r) = D(4a-a) = 3Da

Q

Q

When r = 3a

p(r) = D(4a - 3a) = Da

Q

Q

Total charge = Q

ρ(r) = D(4a-a) = 3Da

Q

_{1}= ρV = (3Da)(4/3πa^{3})Q

_{1}= 4πDa^{4}When r = 3a

p(r) = D(4a - 3a) = Da

Q

_{2}= ρV = (Da)(4/3π(3a)^{3})Q

_{2}= 12πDa^{4}Total charge = Q

_{2}-Q_{1}= 8πDa^{4}Now common sense is screaming at me saying this is wrong, but I am unsure where I should be going if this is the case.

Would love your feedback.

Thanks in advance.