Explain why the gravitational potential at any point outside an isolated, uniform spherical mass is negative, and describe how this potential changes as a small test mass is moved from the surface of the sphere to a very large distance away.

IB DP Physics Higher Level (2023 syllabus) — D.1 Gravitational fields · Explain · 4 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

Model answer (4 marks)

The gravitational potential is defined to be zero at infinity. To move a test mass from any finite distance to infinity an external agent must do positive work, so the potential at that finite distance must be less than zero – i.e. negative. As the test mass is moved away from the surface of the sphere the potential increases (becomes less negative) because the magnitude of the negative potential decreases. The potential approaches zero asymptotically as the distance becomes very large; it varies as –GM/r, so the rate of increase slows with distance.

Examiner tips

  • State the zero‑point definition first, then explain the sign of the potential using work done by an external agent.
  • Show the potential change as the mass moves outward and note the asymptotic approach to zero.
  • Use the correct formula –GM/r to justify the rate of change.
  • Keep the answer concise – 4 marks only.

Common mistakes

  • Assuming the potential is positive at finite distances; forgetting the zero‑point definition.
  • Confusing potential with potential energy and not mentioning the external work required.
  • Lacking the asymptotic behaviour or the 1/r dependence of the potential.

Mark scheme (4 marks)

  1. Gravitational potential is defined as zero at infinity (at an infinite distance from the mass).
  2. Work must be done by an external agent (energy must be supplied) to move the test mass from any finite distance to infinity, so the potential at any finite distance must be less than zero / is negative.
  3. As the test mass is moved away from the surface, the gravitational potential increases (becomes less negative / increases towards zero).
  4. The potential approaches zero asymptotically as the distance becomes very large / tends to infinity, and the rate of increase decreases with distance (potential varies as −1/r so the gradient decreases).

Key terms in this question

gravitational potential · test mass

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