# What is a field (physics)

1. Jul 24, 2014

### 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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