Comparing Fields

01Comparing Fields

Gravitational & electric fields

This section covers the main similarities and differences between gravitational and electric forces, fields, potentials and equipotential surfaces.

Force, field strength and potential

Gravitational and electric fields have closely matching mathematical forms. For point sources, their force and radial field-strength equations contain an inverse-square dependence on distance, whereas their potentials vary inversely with distance.

Property	Gravitational field	Electric field
Source	Mass	Charge
Force magnitude between two point sources	$F=\frac{Gm_1m_2}{r^2}$	$F=\frac{\lvert Q_1Q_2\rvert}{4\pi\varepsilon_0r^2}$
Possible interaction	Only attraction between masses	Attraction for opposite signs and repulsion for matching signs
Field strength magnitude	$g=\frac{F}{m}$	$E=\frac{F}{\lvert Q\rvert}$
Field strength magnitude due to a point source	$g=\frac{GM}{r^2}$	$E=\frac{\lvert Q\rvert}{4\pi\varepsilon_0r^2}$
Potential	$V=-\frac{GM}{r}$	$V=\frac{Q}{4\pi\varepsilon_0r}$
External work with no change in kinetic energy	$\Delta W=m\Delta V$	$\Delta W=Q\Delta V$

In both cases, the point-source force obeys

$F\propto\frac{1}{r^2},$

and the magnitude of radial field strength has the same distance dependence:

$\text{field strength}\propto\frac{1}{r^2}.$

The corresponding radial potentials change in proportion to $1/r$ .

Radial field lines

Point mass

A point mass produces an inward radial field pattern.

Negative point charge

A negative point charge also produces an inward radial pattern. Reversing the sign of the charge reverses the electric-field direction.

A point mass and a negative point charge therefore give the same inward radial line geometry.

Uniform fields and equipotentials

Uniform gravitational field

Straight, parallel and equally separated field lines represent a uniform gravitational field. The dashed equipotentials lie at right angles to the field and are evenly spaced for equal potential steps.

Uniform electric field

Between oppositely charged parallel plates, the central electric field has the same straight and evenly spaced line pattern, with parallel equipotentials perpendicular to it.

For a point source, equipotential surfaces form spheres centred on the mass or charge. For a uniform field, equal potential steps correspond to evenly separated parallel equipotential surfaces.

Important differences

Source property

Mass is the property that couples an object to a gravitational field, whereas electric charge determines its interaction with an electric field.

Direction of interaction

There is no repulsive gravitational interaction between masses. Electric interactions change between attraction and repulsion according to the signs of the charges.

Sign of potential

Gravitational potential remains below zero, while the sign of electric potential depends on whether the source charge is positive or negative.

Between subatomic particles, gravity is much weaker than the electric interaction. For two protons, the electric force is about $1.2\times10^{36}$ times the gravitational force. The comparison depends on the masses and charges; $G$ and the Coulomb constant $k$ have different units and cannot be compared on their own.

Exam Tip: In a comparison answer, organise points in matched pairs: compare the two force laws, then field strength, potential and field patterns. Finish by contrasting the one-way attractive nature of gravity with the two possible directions of electric interaction.