Potential and Field Graphs
For a finite nonsingular spherical density, V(r) is continuous at the surface and g_r=-dV/dr; for a uniform solid sphere the interior potential is quadratic in r and joins -GM/r smoothly at r=R.
Why this shows up in the exam
Selecting potential graphs · Finding zero-field locations · Checking boundary continuity
Learn the idea
The field is minus the slope of V(r), while potential remains continuous across ordinary mass boundaries. A graph encodes both force direction and strength: steep potential means strong field, and a flat potential means zero field.
🧠 Memory hook: Read field from negative slope, not graph height.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- g_r = -dV/dr — radial field from potential-curve slope
- V_uniform(r<=R) = -GM(3R²-r²)/(2R³) — potential inside a uniform solid sphere
How to approach it
- 1Mark centre, surface, and infinity
- 2Check continuity and limiting values
- 3Use negative slope to infer field
Common slip-ups that cost marks
- •Making V discontinuous at a surface
- •Calling zero potential zero field
- •Ignoring the sign of graph slope
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A satellite moves in a circular orbit where GM = 4 x 10^14 m^3/s^2 and orbital radius is 2 x 10^6 m. Find its orbital speed.
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