S3U

Science / Forces / Undergraduate / sfd20

Gravity Inside and Outside a Uniform Sphere

Derive a continuous spherical field and potential using the shell theorem.

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5 questions0m 0s

Prerequisites

Newtonian gravitation, integration, spherical symmetry, and gradients of scalar potentials.

Learn the skill

Let a sphere have total mass M, radius R and constant density. Its enclosed mass at r <= R is M(r/R)^3. The inward field magnitude is GM r/R^3 inside and GM/r^2 outside. With zero potential at infinity, Phi = -GM(3R^2-r^2)/(2R^3) inside and -GM/r outside.

Worked example

At r = R/2, the field magnitude is GM/(2R^2), half the surface value. At the center, the vector field is zero but Phi = -3GM/(2R). Zero field does not require zero potential.

For a spherical source, gravitational field strength outside it is proportional to one divided by squared center distance. Relative values are one at r, one quarter at 2r, and one ninth at 3r.
For a spherical source, gravitational field strength outside it is proportional to one divided by squared center distance. Relative values are one at r, one quarter at 2r, and one ninth at 3r.
Question 1 Which enclosed mass follows from uniform density at r = R/2?
Question 2 The inside field magnitude at r = R/2 is what?
Question 3 What is the potential at the center for this reference?
Question 4 Which statements hold at r = R?
Question 5 At r = 2R, what is the ratio of field magnitude to its surface value?

Further inquiry

Derive both potential expressions by integrating the field and matching at r = R. Check continuity of the first derivative, then compare the second radial derivatives at the density boundary. Explain which conclusions fail if the density is not uniform.

Review criteria

  • Use the enclosed-mass expression only inside the sphere.
  • State the potential reference and enforce matching at the surface.
  • Distinguish a zero field from a zero potential.