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**1. The problem statement, all variables and given/known data**

What is the gravitational potential both inside and outside a spherical shell of inner radius b and outer radius a?

**2. Relevant equations**

φ = ∫

**g⋅da**= -4πGM

_{encl}

g = d∅/dr in the r hat direction

**3. The attempt at a solution**

I can get as far as getting the gravitational field for the three parts of the shell but im not really sure how to determine the limits of integration in order to get the potentials

for (R > a) g*4πR

^{2}= -4πG*(4/3*π(a

^{3}-b

^{3})*ρ)

then ∅ = ∫[G*(4/3*π(a

^{3}-b

^{3})*ρ)]/R

^{2}dr

for (b< R < a) g*4πR

^{2}= -4πG*(4/3*π(R

^{3}-b

^{3})*ρ)

g = 4/3*πρG*(b

^{3}/R

^{2}- R)

∅ = ∫-[4/3*πρG*(b

^{3}/R

^{2}- R)] dr

for (R < b) g = 0 because there is no enclosed mass

and ∅ = constant the constant being determined from the integration limits

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