Gravitational potential (A-level only)

AQA·Physics·3.7.2.3

Gravitational potential and potential difference

Gravitational potential describes the gravitational potential energy at a point per unit mass. If a small mass $$m$$ has gravitational potential energy $$E_{\mathrm{p}}$$ at that point, its gravitational potential $$V$$ is:

V=EpmV = \frac{E_{\mathrm{p}}}{m}

Equivalently, gravitational potential is the work done per unit mass by an external agent in bringing a small mass from infinity to the point without changing its kinetic energy. The mass is taken to be small enough not to significantly alter the gravitational field.

Gravitational potential is conventionally defined as zero at infinity. In an attractive gravitational field, the field does positive work as the mass moves inwards, so the external agent does negative work to prevent the mass gaining kinetic energy. The potential at the point is therefore negative relative to infinity.

Potential is a scalar quantity: it has no direction. Its unit is the joule per kilogram, $$\mathrm{J,kg^{-1}}$$.

Gravitational potential difference

The potential difference between two points is the change in gravitational potential energy per unit mass when moving from the initial point to the final point:

ΔV=Vfinal−Vinitial\Delta V = V_{\mathrm{final}} - V_{\mathrm{initial}}

For a mass $$m$$, multiplying this potential difference by the mass gives its change in gravitational potential energy. If the mass moves without a change in kinetic energy, this is also the work done by the external agent:

ΔEp=Wexternal=mΔV\Delta E_{\mathrm{p}} = W_{\mathrm{external}} = m\Delta V

The work done by the gravitational field has the opposite sign:

Wfield=−mΔVW_{\mathrm{field}} = -m\Delta V

For example, when a mass moves to a higher gravitational potential, $$\Delta V$$ is positive. Its gravitational potential energy increases, the external agent does positive work and the field does negative work. Moving between the same points in the opposite direction reverses these signs.

Gravitational equipotential surfaces

A gravitational equipotential surface is a surface on which gravitational potential has the same value at every point. Gravitational potential is gravitational potential energy per unit mass, so a mass has the same gravitational potential energy wherever it is on that surface.

Moving a mass between two points on the same equipotential surface therefore produces no change in gravitational potential energy. The gravitational field does no net work on the mass between those points. This does not mean that the gravitational force is zero: the force can act on the mass without doing work along the surface.

Gravitational field direction at an equipotential surface

The gravitational field is perpendicular to an equipotential surface. At any point on the surface, it has no component along the surface that could do work on a mass moving along it. A field component along the surface would do work during such a movement, changing the mass’s gravitational potential energy and contradicting the fact that the potential is the same throughout the surface.

Gravitational potential in a radial field

Gravitational potential at a point is gravitational potential energy per unit mass at that point. Outside a spherical mass, its value depends on the distance from the centre of the mass.

Taking the potential to be zero at infinity, the potential at distance $$r$$ from the centre of a spherical mass $$M$$ is:

V=−GMrV=-\frac{GM}{r}

Here $$G$$ is the gravitational constant. The negative sign means that the potential at any finite distance is below its value at infinity. Energy must be supplied to move a mass from that distance to infinity without changing its kinetic energy.

As $$r$$ increases, $$V$$ becomes less negative: it rises towards zero. Its magnitude, $$GM/r$$, decreases in proportion to $$1/r$$.

Gravitational potential and field strength

The gravitational field is directed towards the spherical mass. Its strength has magnitude:

g=GMr2g=\frac{GM}{r^2}

The two quantities change differently with distance:

  • A graph of field-strength magnitude against $$r$$ decreases towards zero as $$1/r^2$$.
  • A graph of potential against $$r$$ rises towards zero from below as $$-1/r$$.

Neither the field-strength magnitude nor the potential reaches zero at a finite distance.

Gravitational field as a potential gradient

Gravitational potential describes how gravitational potential energy per unit mass varies with position. Gravitational field strength describes the force per unit mass. The field component in a chosen direction is the negative gradient of potential in that direction: the field points towards decreasing potential.

Field from a potential–distance graph

Take the radial coordinate $$r$$ to increase outwards. The radial field component $$g_r$$ is positive if the field points outwards and negative if it points inwards. For a constant potential gradient,

gr=−ΔVΔrg_r=-\frac{\Delta V}{\Delta r}

where $$V$$ is gravitational potential. More generally, the field component at a particular position is the negative local gradient:

gr=−dVdrg_r=-\frac{\mathrm{d}V}{\mathrm{d}r}

On a graph of potential against radial distance, find the gradient of the tangent at the position of interest, then reverse its sign. For example, if potential increases outwards, the gradient is positive, so $$g_r$$ is negative: the field points inwards.

Potential change from a field–distance graph

The relationship can also be used in reverse. Between radial positions $$r_1$$ and $$r_2$$, the change in potential is the negative signed area under a graph of $$g_r$$ against $$r$$:

ΔV=V(r2)−V(r1)=−∫r1r2gr dr\Delta V=V(r_2)-V(r_1)=-\int_{r_1}^{r_2}g_r\,\mathrm{d}r

For an outward displacement in an inward field, $$g_r$$ is negative. The signed area is therefore negative, and the potential change is positive.

A graph may instead show the positive magnitude of the inward field, rather than the signed component $$g_r$$. On that graph, the area between two positions gives the increase in potential for an outward displacement.