Field concept

Main Question or Discussion Point

I have been searching the web for a good explanation of the concept of field but I failed to find a good one. Could somebody provide me with a good definition of this important concept of physics? I'm specially interested in its relation with waves.
Thanks.

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I have been searching the web for a good explanation of the concept of field
You have probably found the mathematician's definition, which contains the the idea used in physics and engineering as a subset.

Here is my offering for the physical world.

First think of a region of space with spatial co-ordinates - 2 dimensional (x, y) or 3 dimensional (x y, z).

This is just an (empty ?) region of space.

But let us now introduce some physical objects. Let us say that our physical region of space contains a cup of coffee. Let us concentrate on the coffee and ignore the cup for our region.

Every point in the coffee has a temperature. The temperatures may be all the same or they may vary, but every point has one.

The temperature distribution throughout the coffee is a field (a temperature field).

More formally, a field is a region of space where we can assign a value to some physical quantity of interest at every point. No points are without a value.

Now temperature is a scalar. We say that a temperature field is a scalar field.

We can also stir the coffee. Then we find that we can assign a velocity to the fluid at every point in our region of space.

This is known as a velocity field.

Velocity is a vector and the velocity field is called a vector field. Note that the centre line of the strirred coffee has zero velocity. Zero is a valid vector value to assign to points on this line.

We can also choose more complicated physical objects such as stress to define a stress field.

The important characteristic is that every point has some value (including possibly zero).

Does this help as a beginning ?

Last edited:
HallsofIvy
Homework Helper
A "field" is simply a way of assigning a value to each point in space. A "vector field" in particular, assigns a vector to each point. So, for example, in talking about wind velocity at each point in space, we are talking about a vector field. You can also have "scalar fields" which assign a number to every point. An example of that would be a "temperature field" which assigns to every point the air temperature there. Another would be a "pressure field which would assing to every point the air pressure at that point.

Thanks both for your inputs. It sure helps as a beginning, I moreless had these notions but it's nice to see them well worded.
As I said I would like to understand better how waves could constitute fields (in case they do).

Waves don't constitute fields.

Both myself and HOI have explicitly stated a most important and fundamental condition for a field.

That is takes in, applies to, assigns a value to every point in the spatial extent of the field. Waves can apply to somes points and not others.

But to move on.

The importance of having a value at every point is that we can do mathematical analysis with such a system.
If there is one and only one value assigned at each point then we can create functions that select certain points of the field an not others. (That is what a wave function does)
We can calculate the difference or change in value from one point to another.
Since there are as many points along each axis as there are real numbers we can do calculus with these differences.
We can even calculate derived fields, such as a displacement field.

How are we doing now|?

Waves don't constitute fields.

Both myself and HOI have explicitly stated a most important and fundamental condition for a field.

That is takes in, applies to, assigns a value to every point in the spatial extent of the field. Waves can apply to somes points and not others.

But to move on.

The importance of having a value at every point is that we can do mathematical analysis with such a system.
If there is one and only one value assigned at each point then we can create functions that select certain points of the field an not others. (That is what a wave function does)
We can calculate the difference or change in value from one point to another.
Since there are as many points along each axis as there are real numbers we can do calculus with these differences.
We can even calculate derived fields, such as a displacement field.

How are we doing now|?
Ok, I think that clarifies a lot. Since you mention the wave function, I guess when I said that waves constitute fields I was thinking that in quantum field, its quanta are the particles of the field(with infinite degrees of freedom and that particle would be represented by a wave function and in that sense this wave functions would constitute the quantum field.
I don't know if this picture can be translated to the classic field, or if it can be fitted to your explanation that wave functions select points of the field. Probably not.
Sorry if I convoluted the OP question.

OK next step.

Fields have a 'field variable' ( eg temperature, gravitational potential or gravitational force of attraction) and one two or three spatial variables.

The spatial variables can be of finite extent (eg the coffee), in which case they have a boundary, or they can extend to infinity, in which case they have no boundary.

The field variable can only be infinite at a finite number of points, known as singularities. Otherwise it can be zero or finite throughout, in which case there are no singularities.

A field with no singularities is conservative. This is a very important concept.

Now introduce time. Fields can vary with time, but time is not a basic variable of the field. 'Each snapshot' in time is a complete field. So, although we talk of 'time varying fields', strictly there is a new distinct field at each time point or instant.
This is where waves come in because the wave equation provides the connection between space and time. Note the quantum wave equation is a different animal. By 'wave function' I meant a solution of the wave equation, not a solution of the Schroedinger equation. Such a solution is a function of both space and time.

So waves have a 'field variable' plus one two or three spatial variables, plus a time variable.

Goodnight for today.

Thanks, you're really good at explaining these concepts.

Now I understand better the meaning of EM waves as oscillating (cyclicly varying in time) electromagnetic fields.

More doubts that arise from your answer, I understand that the classical fields, whose better example would be Maxwell's EM fields theory, seem to depend upon a previous background space and time, but if we take the other main "classical" field theory, gravitation, wich is GR,it has no previous geometry or background spacetime ,instead they are created by the very matter-energy fields that rather than the force field of Newton produce curvature. So is GR really a classical field theory,or is only classical in the sense that is "not quantum"?

I also found very interesting your comments about singularities and conservative fields. Maybe you have some insight about this: does the fact that GR finds some issues with the conservation of energy globally have anything to do with its postulated singularities (blackholes, bigbang)?

Not much time here tonight, but here goes.

It is important to sort the sheep from the goats and stick to the definitions. By this I mean to distinguish between the description/specification of the field itself and disturbances or changes in the field. That way you can use all the theorems that are proven for objects that fit those definitions. Disturbances/changes apply only to the field variable. Sometimes, particularly with vector fields, the change itself is a(nother) field. For example the field of displacement vectors derived from a velocity field.
But beware. Choose an origin. Does any set of position vectors, using this origin form a vector field?

For instance a singularity is a point in space where the field variable does not have a derivative for some reason. Often this is because the variable goes off to infinity at this point. Such singularities are called poles by electrical engineers and complex analysts.
You would have to ask an astrophysicist if classing a black hole as a singularity corresponds to this definition.
The importance of singularities are that they form ways by which we can introduce (or extract) quantities of the field variable from the field. They form the sources and sinks.

You cannot use every physical quantity as a field variable. Take for instance energy. Populist writing often refers to 'energy fields'.
And yes, you can have field denoting potential energy in space.
But could you show a kinetic energy field in the same space? Would such have any meaning?
You can have a temperature field, but how about specific heat field?

Yes you can extend the underlying region to four (or more) axes, so long as the axes are considered equivalent, to define a field. This is the basis of spacetime.
However there are many variables you cannot use as field variables in spacetime, that you can use in ordinary space, so have a care Will Robinson.

The importance of singularities are that they form ways by which we can introduce (or extract) quantities of the field variable from the field. They form the sources and sinks.
Can you give me some example of this?

You cannot use every physical quantity as a field variable. Take for instance energy. Populist writing often refers to 'energy fields'.
And yes, you can have field denoting potential energy in space.
But could you show a kinetic energy field in the same space? Would such have any meaning?
I understand not every physical quantity can be a field variable, it has to be a function of position, certainly kinetic energy doesn't qualify.
there are many variables you cannot use as field variables in spacetime, that you can use in ordinary space
I guess those that can't be expressed as tensors, or those that don't vary as a function of time and position together, maybe?