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Force field (physics)

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Originally a term coined by Michael Faraday to provide an intuitive paradigm, but theoretical construct (in the Kuhnian sense), for the behavior of electromagnetic fields, the term force field refers to the lines of force one object (the "source object") exerts on another object or a collection of other objects. An object might be a mass particle or an electric or magnetic charge, for example. The lines do not have to be straight, in the Euclidean geometry case, but may be curved. Faraday called these theoretical connections between objects lines of force because the objects are most directly connected to the source object along this line.

A conservative force field is a special kind of vector field that can be represented as the gradient of a potential.

Note that a force field does not exist in reality, per se, but it is really a Kuhnian construct that allows scientists to visualize the effects of objects on other objects; in other words, it makes the math easy.

A local Newtonian gravitational field near Earth ground typically consists of a uniform array of vectors pointing in one direction---downwards, towards the ground; its force field is represented by the Cartesian vector $\vec{F}=m\cdot g (-\hat{z})$ , where $\hat{z}$ points in a direction away from the ground, and m refers to the mass, and g refers to the acceleration due to gravity.
A global Gravitational field consists of a spherical array of vectors pointing towards the center of gravity. Its classical force field, in spherical coordinates, is represented by the vector, $\vec{F}=\frac{G m M}{r^2}(-\hat{r})$ , which is just Newton's Law of Gravity, with the radial unit $\hat{r}$ vector pointing towards the origin of the sphere (center of the Earth).
A conservative Electric field has an electric charge (or a smeared plum pudding of electric charges) as its source object. In the case of the point charges, the force field is represented by $\vec{F}=\frac{kqQ}{r^2}(-\vec{R})$ , where $\vec{R}$ is the position vector that represents the straightest line between the source charge and the other charge.
A static Magnetic field has a magnetic charge (a magnetic monopole or a charge distribution).
The electromagnetic force is given by the Lorentz force formula, which in SI units is, $\vec{F}=q(\vec{E} + \vec{v} \times \vec{B})$ .