What is the Definition and Function of a Field in Physics?

[Greg Bernhardt]

Definition/Summary

A field is a map that attaches a (scalar, vector, tensor, etc.) value to every element of an underlying space.

For example, the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ are vector fields over three-dimensional space, while the electromagnetic field is the Faraday tensor field $(\mathbf{E};\mathbf{B})$ over four-dimensional space-time.

A field may be a force, the potential of a force, or something ordinary such as temperature.

The force exerted by a force field on a body depends on the strength of the field, and on various characteristic of the body (including mass, velocity, spin, and various types of charge).

The units in which a force field is measured depend on those characteristics (so, for example, the units of $\mathbf{E}$ have dimensions of velocity times the units of $\mathbf{B}$).

Equations

Lorentz force (for electromagnetic field):

$$\mathbf{F}\ =\ q(\mathbf{E}\ +\ \mathbf{v}\times\mathbf{B})$$

Extended explanation

Flux:

The flux of a field through a surface is the total component of its strength perpendicular to that surface.

Conservative vector field:

A vector field is conservative if it is the gradient of a (non-unique) scalar field (the potential):

$$\mathbf{V}\ =\ \nabla\,\phi$$

So the curl of a conservative vector field is zero (the field is irrotational):

$$\nabla\ \times\ \mathbf{V}\ =\ \nabla\ \times\ \nabla\,\phi\ =\ 0$$

Solenoidal vector field:

A vector field is solenoidal if it is the curl of a (non-unique) vector field (the vector potential):

$$\mathbf{V}\ =\ \nabla\,\times\mathbf{A}$$

So the divergence of a solenoidal vector field is zero:

$$\nabla\cdot\mathbf{V}\ =\ \nabla\ \cdot\ \nabla\,\times\mathbf{A}\ =\ 0$$

Any vector field may be expressed as the sum of a conservative vector field and a solenoidal vector field.​

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[AaronS]

A field is a concept that attaches a value (scalar, vector, tensor, etc) to every element of an underlying space. This can be used to represent forces such as electric and magnetic fields, or something more mundane like temperature. The force exerted by a field on a body is determined by the strength of the field and the characteristics of the body. The units used to measure a force field depend on the characteristics. Two equations, Lorentz force (for electromagnetic fields) and flux, are given for further understanding of fields. It is also mentioned that a vector field can either be conservative or solenoidal, which means its curl and divergence respectively will be zero. Lastly, it is stated that any vector field can be expressed as the sum of a conservative and solenoidal vector field.