The gravitational ↔ electric field analogy
Gravitational and electric fields share the same inverse-square structure with mass replaced by charge and G replaced by 1/4πε₀. Both use field = force per unit (mass/charge), potential = work per unit (mass/charge) from infinity, and field = minus potential gradient. Key difference: gravity is only attractive so φ is always negative, while electric forces can attract or repel so V can be either sign.
F = Gm₁m₂/r² ↔ F = Q₁Q₂/4πε₀r² · g = GM/r² ↔ E = Q/4πε₀r² · φ = −GM/r ↔ V = Q/4πε₀r · g = −dφ/dr ↔ E = −dV/drWhat this actually means
Syllabus LO (b) asks you to recognise this analogy in both qualitative and quantitative terms, so it can be examined directly, usually as a compare-and-contrast.
The quantitative map is mechanical. Swap m for Q, swap G for 1/4πε₀, and every gravitational equation becomes its electric twin. Force, field, potential and potential energy all carry over untouched, and so does the inverse-square distance dependence.
The qualitative map is just as tidy. Field lines start on the source and show direction; line density means field strength; equipotentials are perpendicular to field lines; no work is done moving along an equipotential; and field is minus the potential gradient in both.
Now the differences, which is where the marks actually sit. Gravity acts on mass and is ONLY attractive, so g always points inwards and φ is always negative. Electric fields act on charge and can attract or repel, so E can point either way and V can be positive or negative.
Two more contrasts worth having ready. Gravity cannot be shielded, while electric fields can be screened by a conductor. And gravity is fantastically weaker: between two protons the electric repulsion beats the gravitational attraction by about 10³⁶, which is why gravity only matters for astronomical masses.
Practical payoff for revision: learn the gravitation definitions properly now and you get the electric ones nearly free later, since the sentences are word-for-word identical with charge in place of mass.
Claiming gravitational potential can be positive by over-extending the analogy with electric potential.
Prove it — watch it be true
- Display the φ–r and g–r graphs and note φ sits entirely below the axis because gravity only attracts.
- Use the live tangent to confirm g = −dφ/dr, the same gradient relationship that gives E = −dV/dr.
- Compare the 1/r shape of φ with the 1/r² shape of g, the identical pair of shapes you will meet for V and E.