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Physics Dad
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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)πr3
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
Q1 = ρV = (3Da)(4/3πa3)
Q1 = 4πDa4
When r = 3a
p(r) = D(4a - 3a) = Da
Q2 = ρV = (Da)(4/3π(3a)3)
Q2 = 12πDa4
Total charge = Q2-Q1 = 8πDa4
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.