Gravitational Potential due to spherical shell

[Elvis 123456789]

Homework Statement

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

Homework Equations

φ = ∫g⋅da = -4πGM_encl
g = d∅/dr in the r hat direction

The Attempt at a Solution

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

for (R > a) g*4πR²= -4πG*(4/3*π(a³-b³)*ρ)
then ∅ = ∫[G*(4/3*π(a³-b³)*ρ)]/R² dr

for (b< R < a) g*4πR²= -4πG*(4/3*π(R³-b³)*ρ)
g = 4/3*πρG*(b³/R² - R)
∅ = ∫-[4/3*πρG*(b³/R² - R)] dr

for (R < b) g = 0 because there is no enclosed mass
and ∅ = constant the constant being determined from the integration limits

 

[kuruman]

Make 3 drawings, one for each of the three regions. Note that the left side of the equation is always 4πr²g where r is the radius of the sphere that encloses whatever mass. What about the right side? What is an expression for M_encl in each of the 3 regions? That should give you a clue about the limits of integration.

 

[Elvis 123456789]

Make 3 drawings, one for each of the three regions. Note that the left side of the equation is always 4πr²g where r is the radius of the sphere that encloses whatever mass. What about the right side? What is an expression for M_encl in each of the 3 regions? That should give you a clue about the limits of integration.

The expressions for the enclosed mass in each region is in my original post. So I am still not sure how to determine the limits of integration

 

[kuruman]

OK, I now see what you mean by limits of integration. You got M_encl correctly. The problem is with the first equation, φ = ∫g⋅da. I would use r instead of a because a is defined here. The relation between g and φ in spherical symmetry is $g = -\frac{\partial \phi}{\partial r}$ so that $\phi(r) = - \int_{ref}^r g(r) ~dr$. Usually, the lower limit of integration (the reference of potential) is infinity. I hope this clarifies what you need to do.