Example 5.1Worked on the Board! • Find the gravitational potential Φ inside & outside a spherical shell, inner radius b, outer radius a. (Like a similar electrostatic potential problem!) • This is an important & fundamental problem in gravitational theory! If you understand this, you understand the gravitational potential concept!(& probably the electrostatic potential concept as well!) . • Could approach the spherical shell using forces directly (Prob. 5.6), but its MUCH easier with the potential!
Find Φ inside & outside a spherical shell of mass M, mass density ρ,inner radius b & outer radius a. Φ = -G∫[ρ(r)dv/r]. Integrate over V.The difficulty, of course, is properly setting up the integral!If properly set up, doing it is easy!
Summary of ResultsM(4π)ρ(a3 - b3) outside the shell R > a, Φ = -(GM)/R (1) The same as ifM were a point mass at the origin! completely inside the shell R < b, Φ = -2πρG(a2 - b2) (2) Φ = constant, independent of position. within the shell b R a, Φ = -4πρG[a2- (b3/R) - R2] (3) • Also, Φis continuous! If R a, (1) & (3) are the same! If R b, (2) & (3) are the same!
These results arevery important,especially those forR > a, Φ = -(GM)/R This says: The potential at any point outside a spherically symmetric distribution of matter(shell or solid; a solid is composed of infinitesimally thick shells!)is independent of the size of the distribution & is the same as that for a point mass at the origin. To calculate the potential for such a distribution we can consider all mass to be concentrated at the center.
Also, the results are very important for R < b, Φ = -2πρG(a2 - b2) The potential is constant anywhere inside a spherical shell. The force on a test mass m inside the shell is 0!
Given the results for the potential Φ,we can compute the GRAVITATIONAL FIELD g inside, outside & within the spherical shell: g - Φ • Φdepends on R only g is radially directed g = g er = - (dΦ/dR)er [M(4π)ρ(a3 - b3)] outside the shellR > a, g = - (GM)/R2The same as ifM were a point mass at the origin! completely inside the shell R < b, g = 0 Since Φ = constant, independent of position. within the shell b R a, g = (4π)ρG[(b3/R2) - R]
Plots of the potential Φ & the field g inside, outside & within a spherical shell. g - Φ g = - (dΦ/dR) Φ = constant Φ = -(GM)/R g = - (GM)/R2 g = 0