D.1.9 (HL)—Potential gradient

Use the gradient relationship

Gravitational field strength is the negative spatial gradient of gravitational potential. For a graph, use the tangent gradient at the required point; the negative sign makes the field point toward decreasing potential.

Worked example — graph gradient

If a tangent changes by $3.8\times10^8\,\mathrm{J\,kg^{-1}}$ over $4.2\times10^7\,\mathrm{m}$, then $|g|=(3.8\times10^8)/(4.2\times10^7)=9.0\,\mathrm{J\,kg^{-1}m^{-1}}=9.0\,\mathrm{N\,kg^{-1}}$. Direction comes from the negative gradient.

Read a potential–distance graph

The gradient is $\Delta V_g/\Delta r$, with units $\mathrm{J\,kg^{-1}m^{-1}}=\mathrm{N\,kg^{-1}}$. A negative slope gives a positive outward radial magnitude only after the vector direction and sign convention are interpreted. Near a source, the potential changes more rapidly with distance, so the field is stronger.

Connect to work

For a mass $m$ moved between two points, $W=m\Delta V_g$ is the work done on the mass by the external agent under the stated sign convention. The field strength relation is local; potential difference and work compare endpoints.

Common trap

Do not use the average slope over a wide curved section as the field at one point unless the question’s graph is effectively linear there. Do not drop the negative sign without stating whether you are reporting a vector component or a magnitude.