Gravitational Potential due to spherical shell

In summary, the conversation discusses how to determine the gravitational potential both inside and outside a spherical shell with inner radius b and outer radius a. The equations used involve integrating the gravitational field and determining the limits of integration based on the enclosed mass in each region. The conversation also suggests using r instead of a in the first equation for more clarity.
  • #1
Elvis 123456789
158
6

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 = -4πGMencl
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
 
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  • #2
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 Mencl in each of the 3 regions? That should give you a clue about the limits of integration.
 
  • #3
kuruman said:
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 Mencl 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
 
  • #4
OK, I now see what you mean by limits of integration. You got Mencl 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.
 

1. What is gravitational potential due to a spherical shell?

The gravitational potential due to a spherical shell is the potential energy of an object at a certain distance from the center of the shell, caused by the gravitational force of the shell. It is also known as the gravitational potential energy per unit mass.

2. How is the gravitational potential due to a spherical shell calculated?

The gravitational potential due to a spherical shell is calculated using the equation V = -GM/r, where G is the gravitational constant, M is the mass of the shell, and r is the distance from the center of the shell to the object.

3. Does the mass of the spherical shell affect the gravitational potential?

Yes, the mass of the spherical shell directly affects the gravitational potential. As the mass of the shell increases, the gravitational potential also increases.

4. How does the distance from the center of the spherical shell affect the gravitational potential?

The gravitational potential decreases as the distance from the center of the spherical shell increases. This is because the gravitational force decreases with distance according to the inverse-square law.

5. What is the difference between gravitational potential due to a solid sphere and a hollow sphere?

The gravitational potential due to a solid sphere is only dependent on the distance from the center, while the gravitational potential due to a hollow sphere is dependent on both the distance from the center and the thickness of the shell. A solid sphere has a more concentrated mass, resulting in a stronger gravitational potential at the same distance compared to a hollow sphere.

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