Force fields and Newton's law of gravitationAQA A-Level Physics: Revision notes

A force field is a region in which a body experiences a non-contact force. The force is exerted without the bodies touching, and its size and direction depend on position in the field. At A Level you meet three sources of field: mass (gravitational fields), static charge (electric fields) and moving charges (magnetic fields).

A field can be represented by a vector at each point. Its direction must be determined by inspection, for example towards a mass for gravity, or away from a positive charge for electric fields. Field lines show these directions, and lines closer together show a stronger field.

Gravity acts between all matter and is always attractive; there is no gravitational repulsion. Every mass attracts every other mass, but the force is only noticeable when at least one body is very massive, such as a planet or star.

The forces on the two bodies are equal in magnitude and opposite in direction. They are a Newton's third law pair, so a falling apple pulls the Earth upwards with the same force as the Earth pulls the apple down.

The magnitude of the force between two point masses is $F = \dfrac{G m_1 m_2}{r^2}$, where $m_1$ and $m_2$ are the masses, $r$ is the separation of their centres, and $G$ is the gravitational constant, $6.67 \times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}$ (given in the data booklet).

This is an inverse square law: $F \propto 1/r^2$, so doubling the separation divides the force by 4, and trebling it divides the force by 9. The force is proportional to each mass, so doubling one mass doubles the force.

A uniform sphere acts, for points outside it, as if all of its mass were concentrated at its centre. So for planets and stars $r$ is the distance between centres, not between surfaces.

Worked example: Earth ($5.97\times10^{24}$ kg) and Moon ($7.35\times10^{22}$ kg), centres $3.84\times10^{8}$ m apart. $F = \dfrac{6.67\times10^{-11}\times5.97\times10^{24}\times7.35\times10^{22}}{(3.84\times10^{8})^2} = 1.98\times10^{20}$ N.

For a body between two masses the forces act in opposite directions, so subtract them. The resultant is zero at the point where the two forces are equal.

Similarities: both forces obey inverse square laws for point sources; both can be described using field lines; both have the idea of potential and equipotential surfaces; both act at a distance.

Differences: masses always attract, but charges may attract or repel (like charges repel, unlike attract). Electrostatic forces are also vastly larger than gravitational forces between subatomic particles. Moving charges additionally give rise to magnetic forces.