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# Force fields

## Force field definition

Informally, a force field is a region of space in which a non-contact force acts on a body.

This means that an object in a force field experiences a force without physical contact with other objects.

The gravitational field around a planet exerts a force on a satellite orbiting the planet. The gravitational field is a force field.

The idea of fields is explicit in the alignment of iron fillings by a magnetic field.

The two poles of the magnet create a magnetic field that exerts a force on each small piece of iron around it. The iron fillings trace the form of the magnetic field.

The non-contact force varies in magnitude according to the physical properties (e.g. mass, charge, velocity) of the object and the position of the object.

The magnitude of the field at a point in space is known as the field strength. The potential energy of the object at a point in space due to a particular field depends on the field strength at that same point.

## Potential energy in a uniform force field

In a uniform force field (or simply uniform field), the field strength and direction of the field are the same at all points.

While most force fields in the real world are non-uniform, they can be considered to be approximately uniform when considering points which are close together.

The gravitational field along small distances (relative to the size of the earth) near the surface of the earth can be considered to be a uniform field as the distance between the surface of the earth and the atmosphere is small (relative to the radius of the earth).

In a uniform field, the potential energy of an object decreases linearly as the object moves away from the source of the field. This is because all fields extend from a region of higher potential to a region of lower potential.

An object gains potential energy if it moves in the opposite direction of a force field and loses potential energy if it moves in the same direction of the force field.

An object which falls from a certain height loses gravitational potential energy. Equivalently, throwing an object upwards from the ground causes it to gain gravitational potential energy.

## Potential energy equations

The potential energy of an object in a force field is the work that should be done by the field to bring the object from one point to another. In other words, the field "drags" the object.

The force on a particle in a uniform field $F$ is equal to the negative of the gradient of the potential energy of the particle at that point: $$\vecphy{F}=-\frac{dE_{\text{p}}}{d\vecphy{s}}=-\frac{\Delta E_{\text{p}}}{\Delta\vecphy{s}}$$ This implies that the change in potential energy between two points in a uniform field is given by the negative of the product of the force with the change in displacement (i.e. work): $$\Delta E_{\text{p}}=-\vecphy{F}\cdot\Delta\vecphy{s}$$

$E_{\text{p}}=$potential energy; $\vecphy{F}=$force; $\vecphy{s}=$displacement.